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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

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

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

and

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

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

4) Draw the CAST diagram

Which ratios are positive in each quadrant?

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

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)

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

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

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+c22bccosAa2=b2+c22bccosAa^2 = b^2 + c^2 - 2bc cos A or cosA=b2+c2a22bccosA=b2+c2a22bccos A = {b^2 + c^2 - a^2} over 2bc

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

(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°x360°360°x360°-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°θ360°0°θ360°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

Negative angles ( 360°θ0°360°θ0°-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

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 kkk 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: sin1(18)=7,18sin1(18)=7,18sin^{-1}(1 over 8) = 7,18

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

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)

( kkk 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.

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

#### Exercise 2

If cos(θ)=213cos(θ)=213cos(%theta) = -{2 over {sqrt{13}}} and 180°θ360°180°θ360°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)

#### 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.

#### 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(2xy)+tan2(x)sin(2xy)+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)

#### 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)

#### 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}})

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

2) Prove the following identities:

a) 1sin2(x)1+cos(x)=cos(x)1sin2(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)}

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

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

d) 2sin(θ15°)+1=02sin(θ15°)+1=02 sin(%theta - 15^circ) + 1 = 0

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

#### 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°x180°180°x180°-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.

#### Exercise 10

Refer to the diagram:

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

b) Calculate the area of the triangle.

#### Exercise 11

Refer to the diagram.

a) Calculate the length of BC

b) Calculate the size of angle B.

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