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Trigonometry: Revision of grade 11

Module by: Pinelands High School. E-mail the author

Trigonometry Summary

Basic knowledge and skills required for Grade 12

You should be able to:

1) Write down the definitions of the three trigonometric ratios

Soh Cah Toa – ratios in terms of side opposite reference angle, side adjacent to reference angle and hypotenuse (always the right angle)

sin(θ)=opphyp=casin(θ)=opphyp=casin (%theta) = opp over hyp = c over a

cos(θ)=adjhyp=bacos(θ)=adjhyp=bacos (%theta) = adj over hyp = b over a

tan(θ)=oppadj=cbtan(θ)=oppadj=cbtan(%theta) = opp over adj = c over b

Figure 1
Figure 1 (graphics1.png)

2) Sketch and use the special (standard) triangles:

Triangle with angles 30°,60°,90°30°,60°,90°30^{circ}, 60^{circ}, 90^{circ}

Figure 2
Figure 2 (graphics2.png)

and

Triangle with angles 45°,45°,90°45°,45°,90°45^{circ}, 45^{circ}, 90^{circ}

Figure 3
Figure 3 (graphics3.png)

3) Write down the definitions of the three trigonometric ratios and their reciprocals for angles of any size in a Cartesian plane (Syr Cxr Tyx)

Shield your rear, 'Cause x-rays Tan your exterior – ratios in terms of the radius and of the co-ordinates of the points at the end of the radius.

sin(θ)=yrsin(θ)=yrsin(%theta) = y over r

cos(θ)=xrcos(θ)=xrcos(%theta) = x over r

tan(θ)=yxtan(θ)=yxtan(%theta) = y over x

Figure 4
Figure 4 (graphics4.png)

4) Draw the CAST diagram

Which ratios are positive in each quadrant?

Figure 5
Figure 5 (graphics5.png)

5) Distinguish between positive angles (anti-clockwise) and negative angles (clockwise)

Figure 6
Figure 6 (graphics6.png)

6) Note that:

cos(−θ)=cos(θ)cos(−θ)=cos(θ)cos(- %theta) = cos (%theta)

sin(−θ)=−sin(θ)sin(−θ)=−sin(θ)sin(- %theta) = - sin(%theta)

tan(−θ)=−tan(θ)tan(−θ)=−tan(θ)tan(- %theta) = - tan(%theta)

7) Write down and apply the 180 degrees rule ( 180°±θ,360°±θ,etc180°±θ,360°±θ,etc180^{circ} +- %theta, ` 360^{circ} +- %theta `, etc)

(RATIO does NOT change, SIGN may change based on CAST)

a) Identify quadrant

b) Use CAST diagram to determine SIGN

c) Reduce angle to reference angle

Example 1

cos(180°−θ)=−cos(θ)cos(180°−θ)=−cos(θ)cos(180^{circ} - %theta) = - cos(%theta) (Second quadrant, CAST: cos negative) tan(210°)=tan(180°+30°)=tan(30°)tan(210°)=tan(180°+30°)=tan(30°)tan(210^circ) = tan(180^{circ} + 30^{circ}) = tan (30^{circ}) (Third quadrant, CAST: tan positive)

Remember the following when using the 180 degree rule:

- We assume θθ%theta is acute – First quadrant

- 180°−θ180°−θ180^{circ} - %theta - Second quadrant

- 180°+θ180°+θ180^circ + %theta - Third quadrant

- 360°−θ360°−θ360^circ - %theta - Fourth quadrant

Figure 7
Figure 7 (graphics7.png)

8) Write down and apply the 90 degree rule ( 90°±θ90°±θ90^circ +- %theta)

(RATIO changes to CO-ratio; SIGN base on CAST and original ratio)

sine and cosine

a) Identify the quadrant

b) Use CAST digrams to determine SIGN and write it down

c) Change ratio to COratio

d) Reduce angle to reference angle

Example 2

sin(90°−θ)=cos(θ)sin(90°−θ)=cos(θ)sin(90^circ - %theta) = cos(%theta) (First quadrant; CAST: sin positive)

cos(120°)=cos(90°+30°)=−sin(30°)cos(120°)=cos(90°+30°)=−sin(30°)cos(120^circ) = cos(90^circ + 30^circ) = -sin(30^circ) (Second quadrant, CAST: cos negative)

9) Write down the Pythagorean identity (Square identity)

sin 2 ( θ ) + cos 2 ( θ ) = 1 sin 2 ( θ ) + cos 2 ( θ ) = 1 sin^{2}(%theta) + cos^{2}(%theta) = 1
(1)

10) Write down the Quotient identity

tan ( θ ) = sin ( θ ) cos ( θ ) tan ( θ ) = sin ( θ ) cos ( θ ) tan(%theta) = {sin(%theta)} over {cos(%theta)}
(2)

11) Write down the sine rule, cos rule and area rule (for use in triangles)

Sine rule: sinAa=sinBb=sinCcsinAa=sinBb=sinCc{sin A} over a = {sin B} over b = {sin C} over c or asinA=bsinB=csinCasinA=bsinB=csinCa over {sin A} = b over {sin B} = c over {sin C}

Cosine rule: a2=b2+c2−2bccosAa2=b2+c2−2bccosAa^2 = b^2 + c^2 - 2bc cos A or cosA=b2+c2−a22bccosA=b2+c2−a22bccos A = {b^2 + c^2 - a^2} over 2bc

Area rule: AreaΔABC=12absinCAreaΔABC=12absinCArea %DELTA ABC = {1 over 2} ab sin C

Applying your knowledge

(Be aware of the broad categories of questions that may be asked)

You must be able to:

1) Use a sketch to find x, y or r and then ratios and angles in the Cartesian Plane (all four quadrants) using the theorem of Pythagoras ( x2+y2=r2x2+y2=r2x^2 + y^2 = r^2) and the CAST diagram.

2) Determine the ratio of any positive angle, as well as the angles such as 0°,90°,180°0°,90°,180°0^circ , 90^circ , 180^circ, etc, and negative angles in the Cartesian Plane.

3) Determine angles for which ratios are undefined (denominator zero), e.g. tan(90°)tan(90°)tan(90^circ), etc.

4) Simplify trigonometric expressions using the theory described in the knowledge section above.

When simplifying expressions:

a) Use reduction formulae to change all angles to acute angles

b) It is often a good idea to write the expression in terms of sine and cosine as far as possible.

c) Look out for double angles and co-ratios

5) Find domain specific solutions to trigonometric equations: e.g. Solve for x if 2sin(x)=0,432sin(x)=0,432 sin(x) = 0,43 and −360°≤x≤360°−360°≤x≤360°-360^circ leslant x leslant 360^circ

REMEMBER the basic steps for solving a trigonometric equation:

a) Isolate the ratio with the unknown angle

b) Ignore the +/- sign and determine the reference angle

c) Use the +/- sign to determine the quadrant(s) in which the angle(s) must lie

d) Determine your solution(s), using the reference angle.

Remember the following when working out your solution using the reference angle:

Positive angles ( 0°≤θ≤360°0°≤θ≤360°0^circ leslant %theta leslant 360^circ)

First quadrant: θ=refangleθ=refangle%theta = ref `angle

Second quadrant: θ=180°−refangleθ=180°−refangle%theta = 180^circ - ref ` angle

Third quadrant: θ=180°+refangleθ=180°+refangle%theta = 180^circ + ref ` angle

Fourth quadrant: θ=360°−refangleθ=360°−refangle%theta = 360^circ - ref ` angle

Figure 8
Figure 8 (graphics8.png)

Negative angles ( −360°≤θ≤0°−360°≤θ≤0°-360^circ leslant %theta leslant 0^circ)

First quadrant: θ=360°+refangleθ=360°+refangle%theta = 360^circ + ref ` angle

Second quadrant: θ=−180°−refangleθ=−180°−refangle%theta = -180^circ - ref ` angle

Third quadrant: θ=−180°+refangleθ=−180°+refangle%theta = -180^circ + ref ` angle

Fourth quadrant: θ=0°−refangleθ=0°−refangle%theta = 0^circ - ref ` angle

Figure 9
Figure 9 (graphics9.png)

6) Find the GENERAL Solution to trigonometric equations (at step d) above, remember to add 360°k360°k360^{circ} k(for sine and cosine) or 180°k180°k180^{circ}k (for tan) to your answer and to specify that k∈ℤk∈ℤk in setZ)

Example 3

2 sin ( x + 20 ° ) + 1 = 3 4 2 sin ( x + 20 ° ) = − 1 4 sin ( x + 20 ° ) = − 1 8 2 sin ( x + 20 ° ) + 1 = 3 4 2 sin ( x + 20 ° ) = − 1 4 sin ( x + 20 ° ) = − 1 8 2 sin(x + 20^circ) + 1 = 3 over 4 newline 2 sin(x + 20^circ) = -1 over 4 newline sin(x + 20^circ) = -1 over 8
(3)

Ref angle: sin−1(18)=7,18sin−1(18)=7,18sin^{-1}(1 over 8) = 7,18

3rd quadrant:

x + 20 ° = 180 ° + 7,18 + 360 ° k x = 167,18 + 360 ° k x + 20 ° = 180 ° + 7,18 + 360 ° k x = 167,18 + 360 ° k x + 20^circ = 180^circ + 7,18 + 360^{circ} k newline x = 167,18 + 360^{circ}k
(4)

or

4th quadrant:

x + 20 ° = 360 ° − 7,18 + 360 ° k x = 332,82 + 360 ° k x + 20 ° = 360 ° − 7,18 + 360 ° k x = 332,82 + 360 ° k x + 20^circ = 360^circ - 7,18 + 360^{circ} k newline x = 332,82 + 360^{circ}k
(5)

( k∈ℤk∈ℤk in setZ)

7) Solve right-angled triangles (SohCahToa)

8) Use the sine rule, cos rule and area rule in practical applications, i.e. solving triangles, finding angles and side lengths, etc.

Figure 10
Figure 10 (graphics10.png)
9) Solve problems in two dimensions

NB: Make sure that you know how to use your calculator to find angles and ratios

Exercise 1

In which quadrant does θθ%theta lie if:

a) sin(θ)<0sin(θ)<0sin(%theta) < 0 and tan(θ)tan(θ)tan(%theta)

b) cos(θ)>0cos(θ)>0cos(%theta) > 0 and sin(θ)<0sin(θ)<0sin(%theta) < 0

c) tan(θ)>0tan(θ)>0tan(%theta) > 0 and cos(θ)>0cos(θ)>0cos(%theta) > 0

[ Show Solution ]

Exercise 2

If cos(θ)=−213cos(θ)=−213cos(%theta) = -{2 over {sqrt{13}}} and 180°≤θ≤360°180°≤θ≤360°180^circ leslant %theta leslant 360^circ, use a sketch to determine the value of:

a) tan(θ)tan(θ)tan(%theta)

b) sin(θ)cos(θ)sin(θ)cos(θ)sin(%theta) cos(%theta)

[ Show Solution ]

Exercise 3

If tan(θ)=ttan(θ)=ttan(%theta) = t and θθ%theta is acute, determine sin(θ)sin(θ)sin(%theta) in terms of t.

[ Show Solution ]

Exercise 4

1) If x=87,6°x=87,6°x = 87,6^circ and y=240,2°y=240,2°y = 240,2^circ, use a calculator to evaluate each of the following correct to two decimal places:

a) cos(x+y)cos(x+y)cos(x + y)

b) sin(2x−y)+tan2(x)sin(2x−y)+tan2(x)sin(2x - y) + tan^{2}(x)

c) sin(y)cos(x)+3tan(2x)sin(y)cos(x)+3tan(2x){sin(y)} over {cos(x)} + 3 tan(2x)

d) cos(y)2cos(y)2{cos(y)} over 2

2) Without using a calculator, find the value of:

a) tan(310°)sin(60°)tan(310°)sin(60°)tan(310^circ) sin(60^circ)

b) cos2(45°)+sin(30°)cos2(45°)+sin(30°)cos^{2}(45^circ) + sin(30^circ)

c) cos(30°)+sin(60°)cos(30°)+sin(60°)cos(30^circ) + sin(60^circ)

[ Show Solution ]

Exercise 5

Reduce each of the following to a trigonometric ratio of x:

a) sin(180°+x)sin(180°+x)sin(180^circ + x)

b) tan(90°+x)tan(90°+x)tan(90^circ + x)

c) cos(360°−x)cos(360°−x)cos(360^circ - x)

d) cos(90°−x)sin(360°−x)cos(90°−x)sin(360°−x){cos(90^circ - x)} over {sin(360^circ - x)}

e) sin(x)−cos(90°−x)−tan(180°−x)sin(x)−cos(90°−x)−tan(180°−x)sin(x) - cos(90^circ - x) - tan(180^circ - x)

[ Show Solution ]

Exercise 6

1) Evaluate without using a calculator:

a) tan(120°)tan(120°)tan(120^circ)

b) cos(630°)cos(630°)cos(630^circ)

c) sin(150°)+tan(330°)cos(30°)sin(150°)+tan(330°)cos(30°)sin(150^circ) + tan(330^circ) cos(30^circ)

d) tan(315°)+cos(300°)sin(150°)+tan(135°)tan(315°)+cos(300°)sin(150°)+tan(135°){tan(315^circ) + cos(300^circ)} over {sin(150^circ) + tan(135^circ)}

2) Prove that sin(240°)tan(300°)+cos(330°)=12(3+3)sin(240°)tan(300°)+cos(330°)=12(3+3)sin(240^circ) tan(300^circ) + cos(330^circ) = {1 over 2}(3 + {sqrt{3}})

[ Show Solution ]

Exercise 7

1) Use basic trigonometric identities to simplify the following:

a) tan(y)cos(y)tan(y)cos(y)tan(y) cos(y)

b) tan2(y)sin2(y)+tan2(y)cos2(y)tan2(y)sin2(y)+tan2(y)cos2(y)tan^{2}(y) sin^{2}(y) + tan^{2}(y) cos^{2}(y)

c) 1−cos2(y)sin(y)1−cos2(y)sin(y){1 - cos^{2}(y)} over {sin(y)}

2) Prove the following identities:

a) 1−sin2(x)1+cos(x)=cos(x)1−sin2(x)1+cos(x)=cos(x)1 - {sin^{2}(x)} over {1 + cos(x)} = cos(x)

b) cos(x)1+sin(x)+tan(x)=1cos(x)cos(x)1+sin(x)+tan(x)=1cos(x){cos(x)} over {1 + sin(x)} + tan(x) = 1 over {cos(x)}

[ Show Solution ]

Exercise 8

Find the general solution to the following equations. Give answers to two decimal places:

a) sin(θ)=0,515sin(θ)=0,515sin(%theta) = 0,515

b) 3−tan(θ)=2,43−tan(θ)=2,43 - tan(%theta) = 2,4

c) cos(θ+20°)=−0,242cos(θ+20°)=−0,242cos(%theta + 20^circ) = -0,242

d) 2sin(θ−15°)+1=02sin(θ−15°)+1=02 sin(%theta - 15^circ) + 1 = 0

e) cos(2θ)=tan(24°)cos(2θ)=tan(24°)cos(2 %theta) = tan(24^circ)

[ Show Solution ]

Exercise 9

On the same system of axes, draw sketch graphs of:

f(x)=sin(x)f(x)=sin(x)f(x) = sin(x) and g(x)=cos(90°+x)g(x)=cos(90°+x)g(x) = cos(90^circ + x) for the interval −180°≤x≤180°−180°≤x≤180°-180^circ leslant x leslant 180^circ. Use the graphs to answer the following questions:

a) Describe g(x) in terms of a reflection of f(x).

b) Explain why f(x) + g(x) = 0 for all values of x.

[ Show Solution ]

Exercise 10

Refer to the diagram:

Figure 13
Figure 13 (graphics13.png)

a) Calculate the measurement of AB (correct to two decimal places)

b) Calculate the area of the triangle.

[ Show Solution ]

Exercise 11

Refer to the diagram.

Figure 14
Figure 14 (graphics14.png)

a) Calculate the length of BC

b) Calculate the size of angle B.

[ Show Solution ]

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