Everything Science Grade 11 - Everything Maths. Pages·· MB· 7, Downloads. ,2. Hs. (). Co. 58,9. Rh. ,9. Open textbooks offered by Siyavula to anyone wishing to learn maths and science. PDF (CC-BY-ND). Mathematics Grade Read online. Textbooks. English. 11 MATHEMATICS. VERSION 1 CAPS You are allowed and encouraged to copy any of the Everything Maths and Everything Science textbooks. . For off- line reading on your PC, tablet, iPad and Kindle you can download a digital copy.

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You are allowed and encouraged to freely copy this book. You can photocopy, print and distribute it as often as you like. You can download it onto your mobile. Siyavula textbooks: Grade 11 Maths PDF generated: October 29, .. In Grade 10, we worked only with indices that were integers. Siyavula Mathematics Grade 11 Open Textbook - Ebook download as PDF File . pdf), Text File .txt) or read book online. Mathematics VERSION 1 CAPS Everything Maths and Everything Science are trademarks of Siyavula Education.

You can legally photocopy any page or even the entire book. You can download it from www. You can burn it to CD, put on your flash drive, e-mail it around or upload it to your website. The only restriction is that you have to keep this book, its cover, title, contents and short-codes unchanged. This book was derived from the original Free High School Science Texts written by volunteer academics, educators and industry professionals. The Everything Maths and Science series is one of the titles developed and openly released by Siyavula. For more information about the writing and distribution of these or other openly licensed titles:

You can burn it to CD, put on your flash drive, e-mail it around or upload it to your website. The only restriction is that you have to keep this book, its cover, title, contents and short-codes unchanged. This book was derived from the original Free High School Science Texts written by volunteer academics, educators and industry professionals.

The Everything Maths and Science series is one of the titles developed and openly released by Siyavula. For more information about the writing and distribution of these or other openly licensed titles: www. Mark Horner; Dr. Samuel Halliday; Dr. Sarah Blyth; Dr. Rory Adams; Dr. Vanessa Godfrey; Terence Goldberg; Dr.

Sam Halliday; Asheena Hanuman; Dr. William P. Heal; Pierre van Heerden; Dr. Fritha Hennessy; Dr. Benne Holwerda; Dr. Jannie Leach; Nkoana Lebaka; Dr. Marco van Leeuwen; Dr. Matina J. Check your solutions by also solving algebraically. We can solve algebraically to check. We also note that this sytem of equations has the followig restrictions: The solutions to the system of simultaneous equations are 3. We can solve algebraically to check and to get a more accurate answer.

Doing this gives: From the diagram we see that the graphs intersect at approximately 4. If the length is twice the breadth. Note that the breadth cannot be a negative number and so we do not consider this solution. Word problems. Since the length is twice the breadth we can express the length in terms of the breadth: Tsilatsila builds a fence around his rectangular vegetable garden of 8 m2.

We now have the following: We let the length be l and the breadth be b. Kevin has played a few games of ten-pin bowling. If he wants an average score of Now we note the following: Kevin scored 80 more than in the second game.

In the third game. The table below lists the times that Sheila takes to walk the given distances. When an object is dropped or thrown downward. Distance is measured in meters and time is measured in seconds. We are given the distance that the ball falls and the initial velocity so we can solve for t: Then use the equation to answer the following questions: If Sheila were to walk half as fast as she is currently walking.

Find the equation that describes the relationship between time and distance. The graph will be steeper and lie closer to the y -axis. The ends are right-angled triangles having sides 3x. The total surface area of the block is cm2. The length of the block is y.

This is the turning point of the parabola. A wooden block is made as shown in the diagram. Give your answer correct to two decimal places.

Solve for x: Calculate the value of p. Solve for x in terms of p by completing the square: Find one set of 3 7. Saskia and Sven copy it down incorrectly. Determine the correct equation that was on the board. Next we note that for the fraction to equal 0 the numerator must equal to 0.

This gives: Abdoul got stuck along the way. His attempt is shown below: Next equate the two equations and solve for b: These are the e We make a the subject of each equation: Solve the following systems of equations graphically: Substituting this in we get: Now deduce that the total cost.

After doing some research. A stone is thrown vertically upwards and its height in metres above the ground at time t in seconds is given by: Solve the following quadratic equations by either factorisation. Now we can solve for k and the other root. Determine the value of k and the other root. We have solved this and so we can use the solution to solve for y. We have solved this and so we can use the solution to solve for p.

Solve for t: Solve for y: The general term is given for each sequence below. Linear sequences 1. Calculate the missing terms. Write down the next three terms in each of the following sequences: T15 and T Number patterns Calculate how many desks are in the ninth row.

Quadratic sequences 1. Determine the second difference between the terms for the following sequences: Number 1: Quadratic sequences. Quadratic sequences.. Exercise 3 — 3: Calculate the common second difference for each of the following quadratic sequences: Determine whether each of the following sequences is: Second difference: First differences: Given the pattern: Chapter 3. For each of the following patterns. For each of the following sequences.

Is he correct? Given 3. Is she correct? Given the following sequence: Given the following pattern of blocks: There are 15 schools competing in the U16 girls hockey championship and every team must play two matches — one home match and one away match.

A quadratic sequence has a second term equal to 1. Challenge question: Revision 1. Determine the length of the line segment between the following points: Given Q 4. Determine the gradient of the line AB if: Analytical geometry Prove that the line P Q.

Determine the values of x and y. Chapter 4. Calculate the coordinates of the mid-point P x. Draw a sketch and determine the coordinates of N x. Explain your answer.

The two-point form of the straight line equation Determine the equation of the straight line passing through the points: Equation of a line. Gradient—point form of a straight line equation Determine the equation of the straight line: The gradient—intercept form of a straight line equation Determine the equation of the straight line: Multiply by 2: Determine the gradient correct to 1 decimal place of each of the following straight lines.

Angle of inclination 1. Inclination of a line. Determine the angle of inclination correct to 1 decimal place for each of the following: Determine the angle of inclination for each of the following: Inclination of a straight line 1. Parallel lines.

Determine whether or not the following two lines are parallel: Parallel lines 1. Determine the equation of the straight line that passes through the point 1.

Perpendicular lines 1. Calculate whether or not the following two lines are perpendicular: Perpendicular lines. Determine the equation of the straight line that passes through the point 3. Determine the equation of the straight line that passes through the point 2. Determine the equation of the line: Determine the angle of inclination of the following lines: Determine the following: P R intersects the x-axis at S.

The following points are given: Given points S 2. F GH is an isosceles triangle. Quadratic functions.

On separate axes. Chapter 5. Functions Domain and range Give the domain and range for each of the following functions: Intercepts Determine the x. Turning points Determine the turning point of each of the following: Write down the equation of a parabola where the y -axis is the axis of symmetry.

Axis of symmetry 1. Determine the axis of symmetry of each of the following: Sketching parabolas 1. Sketch graphs of the following functions and determine: Draw the following graphs on the same system of axes: Draw a sketch of each of the following graphs: Finding the equation Determine the equations of the following graphs.

Average gradient. Consider the following hyperbolic functions: Hyperbolic functions. Domain and range Determine the domain and range for each of the following functions: Exercise 5 — Asymptotes Determine the asymptotes for each of the following functions: Complete the following for f x and g x: Compare f x and g x and also their axes of symmetry. What do you notice?

Draw the graphs of the following functions and indicate: Illustrate this on your graph. On separate axes, accurately draw each of the following functions: Exponential functions. Asymptote Give the asymptote for each of the following functions: Mixed exercises 1. Label this function as j x. On the same system of axes.

On separate systems of axes. For the diagrams shown below. Axes of symmetry: Find the equation for each of the functions shown below: For each function also determine the following: Revision On separate axes. The sine function. For each function. The sine function 1. Sketch the following graphs on separate axes: For each function in the previous problem determine the following: The cosine function.

For each graph determine: Sketching cosine graphs Exercise 5 — The cosine function 1. Determine the value of a. Two girls are given the following graph: Determine the value of p. For each function determine the following: The tangent function. The tangent function 1.

Determine the equation for each of the following: Indicate the turning points and intercepts on the diagram. Give one of the solutions. The coordinates of the turning point are 3. Suppose 6. What is the new equation then? Give the equation of the new graph originating if: What are the coordinates of the turning point of the shifted parabola? Match each function to the correct column.

Column A: Column B: The columns in the table below give the y -values for the following functions: Column C: Using your knowledge of the effects of p and k draw a rough sketch of the following graphs without a table of values.

Learners must be able to apply the formulae with different notations and in different contexts. Determine the following angles correct to one decimal place: Trigonometry The perpendicular line 3. In ABC. Simplify the following without using a calculator: Let the shorter diagonal be 2x: Given the diagram below.

Chapter 6. Determine the following without using a calculator: The height of an open window is 9 m from the ground. Will the ladder reach the window?

Reduce the following to one trigonometric ratio: Prove the following identities and state restrictions where appropriate: Deriving reduction formulae Exercise 6 — 3: Determine the value of the following expressions without using a calculator: Write the following in terms of a single trigonometric ratio: Reduction formula.

Co-functions 1. Write the following as a function of an acute angle: Exercise 6 — 6: Reduction formulae 1. Write A and B as a single trigonometric ratio: Determine the value of the following. Prove that the following identity is true and state any restrictions: Solving trigonometric equations 1. Trigonometric equations. In the third quadrant: General solution 1.

II quad: III quad: Find the general solution for each equation. Find the general solution for each of the following equations: IV quad: Determine the general solution for each of the following: The area rule 1. Find all the unknown sides and angles of the following triangles: Find the area of Solution: Sine rule 1.

Find the lengths of the sides 2. A AB and BC. In ABD. Find the length of the side ST. Can you determine BC? K Solution: Find the length of the side m. Determine BC. R Solution: A Solution: Solve the following triangles that is. The cosine rule Exercise 6 — The cosine rule 1. Find the length of the third side of the XY Z where: Y Solution: Q is a ship at a point 10 km due south of another ship P.

Determine the largest angle in: At this point the plane is km from Johannesburg. The direct distance between Cape Town and Johannesburg airports is km. W XY Z is a trapezium. From a point Y which is equidistant from X and Z. A a x B c D Solution: A surveyor is trying to determine the distance between points X and Z. However the distance cannot be determined directly as a ridge lies between the two points. Find the area of the shaded triangle in terms of x. Find the area of W XY Z to two decimal places: Write the following as a single trigonometric ratio: Determine the value of the following expression without using a calculator: Prove the following identities: Without the use of a calculator, determine: Find the general solution for the following equations: Given the equation: A is the highest point of a vertical tower AT.

At point N on the tower, n metres from the top of the tower, a bird has made its nest. Collins wants to pave his trapezium-shaped backyard, ABCD. Banele also uses diagonals of length 60 cm and 1 m. Which do you think is the better design? Motivate your answer.

The intersection of the two diagonals cuts the longer diagonal in the ratio 1: Area of a polygon 1. Vuyo decides to make his kite with one diagonal 1 m long and the other diagonal 60 cm long. Area of a polygon. Vuyo and Banele are having a competition to see who can build the best kite using balsa wood a lightweight wood and paper. O is the centre of the bigger semi-circle with a radius of 10 units. Two smaller semi-circles are inscribed into the bigger one. O Chapter 7. Measurement She notices that the dimensions of her desk are in the same proportion as the dimensions of her textbook.

Calculating surface area 1. A popular chocolate container is an equilateral right triangular prism with sides of 34 mm. The box is mm long. Chapter 7. Calculate the surface area of the box to the nearest square centimetre. He discovers a full 2 tin of green paint in his garage and decides to paint the tank not the base. Dimensions of the tank: Right prisms and cylinders. Gordon downloads a cylindrical water tank to catch rain water off his roof. If he uses ml to cover 1 m2.

Volume of prisms and cylinders Exercise 7 — 3: Calculating volume 1. The diameter of the tank is cm and the height is 2. The length of a side of a hexagonal sweet tin is 8 cm and its height is equal to half of the side length. A cylindrical water tank is positioned next to the house so that the rain on the roof runs into the tank.

Finding surface area and volume 1. R is the length from the tip of the cone to its perimeter. Calculate the length of arc P. Determine the length of arc M. An ice-cream cone has a diameter of Determine the value of R. Right pyramids. The effects of k 1. The municipality intends building a swimming pool of volume W 3 cubic metres. Determine the new dimensions of the pool in terms of W remember that the pool must be a cube.

Express b The dimensions of the pool are reduced so that the volume of the pool decreases by a factor of 0. Complete the following sentences: Note that this is not a pentagonal prism since not all the angles are the same size. Which of the following is a net of a cube? Which has the greater volume and which requires the most paper to make? ABCD is a rhombus with sides of length 3 2 x millimetres.

Give a mathematical name for the shape of the box. Determine how much paper is needed to make a box of width 16 cm. Express the area of ABCD in terms of x. Rectangular prism iii. Calculate the volume of the box. The vertical height of the pyramid is 45 cm.

The diagram shows a rectangular pyramid with a base of length 80 cm and breadth 60 cm. A litre of washing powder goes into a standard cubic container at the factory. A cube has sides of length k units.

Pythagoras Pythagoras 7. Is it possible for the children to lose their ball down the pipe? Show your calculations. The soccer ball has a capacity of cc cubic centimetres. Determine x. Perpendicular line from center bisects chord 1. Circle geometry. O x P 4 Q R Solution: In the circle with centre O. Determine T U.

Euclidean geometry In QT O. Pythagoras In Pythagoras 5. Angle at the centre of circle is twice angle at circumference Given O is the centre of the circle. Chapter 8. B and C: Consider the given points A. Alternative method: Exercise 4 — 1: Revision 1.

Prove that the line P Q. Step 8: Determine the gradient of the line AB if: Calculate the coordinates of the mid-point P x. Given Q 4. Determine the values of x and y. Determine the length of the line segment between the following points: The different forms are used depending on the information provided in the problem: Explain your answer. Draw a sketch and determine the coordinates of N x. Given any two points x1. Equation of a line. The two-point form of the straight line equation Determine the equation of the straight line passing through the points: If we are given two points on a straight line.

Exercise 4 — 3: Gradient—point form of a straight line equation Determine the equation of the straight line: Worked example 6: We solve for the two unknowns m and c using simultaneous equations — using the methods of substitution or elimination. Exercise 4 — 4: The gradient—intercept form of a straight line equation Determine the equation of the straight line: Inclination of a line. This is called the angle of inclination of a straight line. We know that gradient is the ratio of a change in the y -direction to a change in the x-direction: We notice that if the gradient changes.

If we are calculating the angle of inclination for a line with a negative gradient. Determine the gradient correct to 1 decimal place of each of the following straight lines. Determine the angle of inclination correct to 1 decimal place for each of the following: Angle of inclination 1. Draw a sketch y 2 2. Worked example 8: Worked example 9: Determine the gradient and angle of inclination of the line through M and N Chapter 4. Exercise 4 — 6: Inclination of a straight line 1.

Determine the angle of inclination for each of the following: What do you notice about mP Q and mRS? Parallel lines. Parallel lines 1. Complete the sentence: Determine the equations of the straight lines P Q and RS. Another method of determining the equation of a straight line is to be given a point on the unknown line. Write the equation in gradient—intercept form We write the given equation in gradient—intercept form and determine the value of m. Determine the equation of the line CD which passes through the point C 2.

Determine whether or not the following two lines are parallel: Determine the equation of the straight line that passes through the point 1. Exercise 4 — 7: Describe the relationship between the lines AB and CD. If not. Perpendicular lines 1. Perpendicular lines. Determine the equation of the straight line AB and the line CD. What do you notice about these products? Deriving the formula: Write the equation in standard form Let the gradient of the unknown line be m1 and the given gradient be m2.

We write the given equation in gradient—intercept form and determine the value of m2. Determine the unknown gradient Since we are given that the two lines are perpendicular. Use the given angle of inclination to determine gradient Let the gradient of the unknown line be m1 and let the given gradient be m2. Calculate whether or not the following two lines are perpendicular: Determine the equation of the straight line that passes through the point 2. Determine the equation of the straight line that passes through the point 3.

Determine the angle of inclination of the following lines: P R intersects the x-axis at S. Determine the following: Exercise 4 — 9: Determine the equation of the line: Consider the sketch above. The following points are given: Given points S 2. F GH is an isosceles triangle.

Functions can be one-to-one relations or many-to-one relations. Functions allow us to visualise relationships in the form of graphs. As a gets closer to 0. As the value of a becomes smaller. The turning point of f x is above the x-axis. Quadratic functions. Every element in the domain maps to only one element in the range. A many-to-one relation associates two or more values of the independent variable with a single value of the dependent variable.

As the value of a becomes larger. The turning point of f x is below the x-axis. Functions Exercise 5 — 1: On separate axes. Consider the three functions given below and answer the questions that follow: The effects of a.

On the same system of axes. The effect of q is a vertical shift. Discuss the similarities and differences. The value of a affects the shape of the graph.

Describe any differences. The range of f x depends on whether the value for a is positive or negative. Determine the range The range of g x can be calculated from: Every point on the y -axis has an x-coordinate of 0. The x-intercept: Every point on the x-axis has a y -coordinate of 0. Exercise 5 — 2: Domain and range Give the domain and range for each of the following functions: Chapter 5.

Exercise 5 — 3: Intercepts Determine the x. Alternative form for quadratic equations: The turning point is p. The minimum value of f x is q.

The maximum value of f x is q. Exercise 5 — 4: Turning points Determine the turning point of each of the following: Axis of symmetry 1. Determine the axis of symmetry of each of the following: Write down the equation of a parabola where the y -axis is the axis of symmetry. Mark the intercepts. State the domain and range of the function. Give the domain and range of the function. Determine the intercepts. State the domain and range Domain: This gives the points 1.

Step 1: Use calculations and sketches to help explain your reasoning. Discuss the two different answers and decide which one is correct. Shifting the equation of a parabola Carl and Eric are doing their Mathematics homework and decide to check each others answers. Work together in pairs. A shift to the right means moving in the positive x direction. Homework question: Determine the new equation of the shifted parabola 1.

If the parabola is shifted 1 unit to the right. Writing an equation of a shifted parabola The parabola is shifted horizontally: The parabola is shifted vertically: The parabola is shifted 3 units down. The parabola is shifted 1 unit to the right.

If the parabola is shifted 3 units down. Draw a sketch of each of the following graphs: Sketch graphs of the following functions and determine: Draw the following graphs on the same system of axes: Exercise 5 — 6: Sketching parabolas 1. Finding the equation of a parabola from the graph If the intercepts are given. Exercise 5 — 7: Finding the equation Determine the equations of the following graphs.

The average gradient between any two points on a curve is the gradient of the straight line passing through the two points. Average gradient.

A 5. What happens to the average gradient as A moves away from B?

The gradient at a point on a curve is the gradient of the tangent to the curve at the given point. At the point where A and C overlap. What happens to the average gradient as A moves towards B?

This line is known as a tangent to the curve. What is the average gradient when A overlaps with B? Find the average gradient between two points P a. Determine the average gradient between P 2. Explain what happens to the average gradient if Q moves closer to P.

Given a curve f x with two points P and Q with P a. We can write the equation for average gradient in another form. Calculate the average gradient between P 2. When the point Q overlaps with the point P. The average gradient between P and Q is: Determine the gradient of the curve at point A. Draw a sketch of the function and determine the average gradient between the points A. Exercise 5 — 9: Consider the following hyperbolic functions: Hyperbolic functions.

Complete the table to summarise the properties of the hyperbolic function: The value of a affects the shape of the graph and its position on the Cartesian plane. The value of q also affects the horizontal asymptotes. The value of p also affects the vertical asymptote.

Discovering the characteristics For functions of the general form: Exercise 5 — Domain and range Determine the domain and range for each of the following functions: Asymptotes Determine the asymptotes for each of the following functions: The asymptotes indicate the values of x for which the function does not exist. For the standard and shifted hyperbolic function.

Complete the following for f x and g x: Compare f x and g x and also their axes of symmetry. Axes of symmetry 1. What do you notice? In order to sketch graphs of functions of the form. The vertical asymptote is also shifted 2 units to the right.

Determine the domain and range Domain: Determine the value of a To determine the value of a we substitute a point on the graph.

Examine the graph and deduce the sign of a We notice that the graph lies in the second and fourth quadrants. Sketching graphs 1. Illustrate this on your graph. Draw the graphs of the following functions and indicate: Exponential functions.

The value of a affects the shape of the graph and its position relative to the horizontal asymptote. Determine the asymptote The asymptote of g x can be calculated as: Asymptote Give the asymptote for each of the following functions: Mark the intercept s and asymptote. We need to solve for p and q.

Use the x-intercept to determine p Substitute 2. Mixed exercises 1. Label this function as j x. For the diagrams shown below. On separate systems of axes. Find the equation for each of the functions shown below: The sine function. Revision On separate axes. For each function also determine the following: Use your sketches of the functions above to complete the following table: The effects of k on a sine graph 1.

If k is negative. Negative angles: For each function determine the following: For each function. To draw a graph of the above equation. Sketch the following graphs on separate axes: The sine function 1. The cosine function. For each function in the previous problem determine the following: The effects of k on a cosine graph 1. Calculating the period: For each graph determine: The effects of p on a cosine graph 1. Determine the minimum turning point At the minimum turning point.

The cosine function 1. Determine the value of p. Determine the value of a. Two girls are given the following graph: The tangent function. The effects of p on a tangent graph 1. Determine the asymptotes The standard tangent graph. The tangent function 1. Determine the equation for each of the following: Give one of the solutions. Indicate the turning points and intercepts on the diagram. Hyperbolic functions: Standard form: Exponential functions: Average gradient: Parabolic functions: Tangent functions: Shifted form: Cosine functions: Sine functions: What is the new equation then?

Give the equation of the new graph originating if: The coordinates of the turning point are 3. Suppose the hyperbola is shifted 3 units to the right and 1 unit down. What are the coordinates of the turning point of the shifted parabola? The columns in the table below give the y -values for the following functions: Match each function to the correct column. Using your knowledge of the effects of p and k draw a rough sketch of the following graphs without a table of values. Trigonometry Solving equations Worked example 1: Identify the opposite and adjacent sides and the hypotenuse Step 2: Determine the value of b Always try to use the information that is given for calculations and not answers that you have worked out in case you have made an error.

Finding an angle Worked example 2: In the calculation below. Determine the angle of inclination of the string correct to one decimal place. Determine the following angles correct to one decimal place: In ABC. Chapter 6. Kite 22 The perpendicular 3. Given the diagram below. Will the ladder reach the window? Simplify the following without using a calculator: The height of an open window is 9 m from the ground. This enables us to solve equations and also to prove other identities.

Quotient identity Investigation: Quotient identity 1. Examine the last two rows of the table and make a conjecture. Complete the table without using a calculator. Trigonometric identities allow us to simplify a given expression so that it contains sine and cosine ratios only. Square identity 1. Draw a sketch and prove your conjecture in general terms. Make a conjecture. Trigonometric identities. Use a calculator to complete the following table: Here are some useful tips for proving identities: Write the expression in terms of sine and cosine only We use the square and quotient identities to write the given expression in terms of sine and cosine and then simplify as far as possible.

Note restrictions When working with fractions. Simplify the left-hand side This is not an equation that needs to be solved. We are required to show that one side of the equation is equal to the other. We can choose either of the two sides to simplify. Reduce the following to one trigonometric ratio: Prove the following identities and state restrictions where appropriate: Exercise 6 — 2: Trigonometric identities 1.

Deriving reduction formulae Investigation: Complete the following reduction formulae: Reduction formula. Determine the value of the following expressions without using a calculator: Write the following in terms of a single trigonometric ratio: We can therefore measure negative angles by rotating in a clockwise direction. From working with functions.

The periodicity of the trigonometric graphs shows this clearly. If k is any integer. Simplify the expression using reduction formulae and special angles 6. Using reduction formula 1. Sine and cosine are known as co-functions. A and B are complementary angles. The function value of an angle is equal to the co-function of its complement.

Simplify the expression using reduction formulae and co-functions Chapter 6. Use the CAST diagram to check in which quadrants the trigonometric ratios are positive and negative. Co-functions 1. Reduction formulae 1. For convenience. Reduction formulae and co-functions: Write A and B as a single trigonometric ratio: Prove that the following identity is true and state any restrictions: Determine the value of the following.

Write the following as a function of an acute angle: Trigonometric equations. Using reduction formulae. If no interval is given. The periodic nature of trigonometric functions means that there are many values that satisfy a given equation. Determine the reference angle use a positive value.

Exercise 6 — 7: Solving trigonometric equations 1. If we do not restrict the solution. Use the CAST diagram to determine where the function is positive or negative depending on the given equation. Check answers using a calculator. This solution is also correct. This solution is correct. In the second quadrant: Make a substitution To solve this equation.

This is because of the periodic nature of the tangent function. Therefore we need only determine the solution: In the third quadrant: Use the CAST diagram to determine the correct quadrants Since the original equation equates a sine and cosine function.

Find the general solution for each equation. Exercise 6 — 8: General solution 1. Find the general solution for each of the following equations: Determine the general solution for each of the following: Exercise 6 — 9: The area rule Investigation: The area rule 1. Consider ABC: Use your results to write a general formula for determining the area of P r q Q p R Chapter 6.