Mathematics Revision

Gr 11 Trigonometry

Trigonometric Ratios

r

yA sin

r

xA cos

x

yA tan

y- axis

x- axis

P (x:y)

A

r

CAST Diagram

ALL

COS

SIN

TAN

Example 1

A

(-4; 3)

y

x

Work out sinA, cosA and tanA

Example 2

If and , evaluate:

1.

2.

03sin5 0tan

cossin

cos5tan4

Example 3

If where and

Determine, using a diagram, an expressionin terms of p for:

1.

2.

5cos

p 0p ]360;180[

tan

sin

cos

Steps to help you

• Make sure your ratio is in ‘simple form’,

i.e.

• Check in which quadrant the point P lies

• Draw a sketch

• Calculate unknown value (x, y, or r) using Pythagoras

• Substitute values into problem and solve

fractionsin

Special Angles

You must be able to calculate the values of ratios of the following angles without using a calculator:

0º, 30º, 45º, 60º, 90º, 180º, 270º, 360º

I remember them by using special triangles,

and my knowledge of graphs.

0º, 90º, 180º, 270º, 360º

30º, 45º, 60º

Special Triangles:

LEARN THESE!

Trig ratios (Using Triangles)

hyp

oppA sin

hyp

adjA cos

adj

oppA tanA

Hypotenuse

Adjacent

Opp

osite

Special Angles: Quick Test

Find the values of the following (no calculator)

)30sin45(cos30tan

)30(cos

60sin2

0cos270sin0sin

0cos90sin

45tan2

60cos30sin

2

2

12

234

3

34

2

2

2

1

Reduction Formulae 1:

Angles bigger than 360º:

sin(360º + A) = sinA

cos(360º + A) = cosA

tan(360º + A) = tanA

Reduction Formulae 2:

Negative Angles:

tan)tan(

cos)cos(

sin)sin(

x

yr

xr

y

Ө

Reduction Formulae 3:

(180º - Ө), (180º + Ө), (360º - Ө)

http://www.mindset.co.za/resources/0000013429/0000018906/0000033287/Reduction.htm

Reduction Formulae 3:

• (90º ± Ө)

Ө

(x,y)

sin)90cos(

cos)90sin(

r

yr

x

sin)90cos(

cos)90sin(

r

yr

x

sin and cos are co-functions

Reduction Formulae Summary:

Sin All

(180º - Ө) Ө (Ө ± 360º)

(90º + Ө) (90º - Ө)

Tan Cos

(180º + Ө) (360º - Ө)

- Ө

Examples:

2

3

225cos.104sin

)135sin(.14cos.300sin.480tan : thatProve

x)-x).sin(360-sin(180tan45

x)-cos(901 :Simplify

)tan().90sin().360cos(

)90sin().180tan().450sin( :Simplify

.sin300tan150

1330cos :Simplify

135tan150sin

)tan(-315-cos300 :Simplify

2

xxx

xxx

Trig Identities:

Quotient Identity:

Proof:

Squares Identity:

Proof:

tancos

sin

1cossin 22

y

x

r

Identities examples:

Prove the following identities:

1.

2.

3.

4.

5.

tansin

cos

cossin

sin21

sin

1

sin

cos

cos1

sincos

1costansin

1)tan1)(sin1(

tansin1

cos1

2

22

22

2

xx

x

x

xx

xxx

Calculator Work:NB: Always check that your calculator is on DEGREES.

1. sin 73° = 0.956304756…2. tan56° = 1,482560969…3. cos320° = 0,766044443…4. 2 sin20° = 2 x sin20° = 0,684040286…5. 2 sin30° + 3 cos10° = 3,954423259…6. tan (21° + 36°) = 1,539864964…7. tan21° + tan260° = 6,055145855…8. sin197°cos13° = -0,284878236… 9. sin2200° = 0,116977778…

Calculator Work: Equations

• sinθ = 0,823

θ = 55,386…

θ = 55,4°

• cosθ = 0,5

θ = 60°

• tanθ = 1,6003

θ = 57,9994…

θ = 58°

• 2 cos θ = 1,264

cos θ = 0,632

θ = 50,8°

Using your calculator to find the reference angle:

Equations

Consider the first example on the previous page sinθ = 0,823

Where can you find the solution off the graph?

Can you see that there are multiple solutions for θ

General vs. Specific Solution

• The general solution gives all the possible solutions for θ.

Remember:• sinθ and cosθ repeat every 360º (+ k.360º)• tanθ repeats every 180º (+ k.180º)

• The specific solution gives solutions for a restricted section.

• E.g. θ Є [-180; 180]

Equations examples

• sinθ = 0,823

ref L = 55,4°

• cosθ = 0,5

ref L = 60°

• tanθ = -1,6003

ref L = 58°

• -2 cos θ = 1,264

cos θ = -0,632

ref L = 50,8°

Find the general solutions for the following equations:

More complex equations eg.s:

Without using a calculator:1.

2.

More advanced:

1. and

2. with

2sin

12

1cos

x

]180;180[

32cos2 02tan

583,1)15cos(3 ]180;180[

Steps to help you

• Find the reference angle (Always +)

• Consider where the solution can be (Quad)

• Manipulate reference angle so that it satisfies given info

• Remember to add k.360º (sin & cos) or k.180º (tan).

• For specific solutions:

Choose answers that satisfy restriction.

sin, cos and area rules

• Sine Rule: (Given: 1 side and 2 angles or 2 sides and 1 angle)

• Cosine Rule: (Given 3 sides, or 2 sides and included angle)

• Area Rule: (Need 2 sides and included angle)

c

C

ba

A sinsinsin

B

Abccba cos2222 bc

acbA

2cos

222

CabArea sin2

1

Example 1: (sin rule)

Calculate c.

A

CB

76º

b = 5,5 cm

Example 2: (cos rule)

Calculate the magnitude of LEDF.

D

FE

9 cm

15 cm

8 cm

Example 3: cos rule

Calculate the length of PR.

P R

Q

120º

5 cm 3 cm

Example 4: area rule

Find the area of Δ XYZ:

45º

12 m

14 m

Extra examples:

cos and sine rule egs.pdf

gr 11 text book sincosarea.pdf

Trig Functions

Terms to know/understand:•Period - wavelength: time it

takes for the wave to complete a cycle

•Amplitude - displacement of the wave: i.e. the height the waves from its centre

•Asymptote - dotted line showing where the function may not touch

•Domain - x-values of function

•Range - y-values of function

Transformations on functions:

qpxy )sin(

)sin(kxay

Translation

Enlargement

Reflection

+: Moves up -: Moves down

+: Moves left -: Moves right

Stretches y-valuesi.e. changes amplitude

Stretches x-valuesi.e. changes period

k

360Period

a<0 Reflects over x-axis