10. Trigonometric Identities and Equations

Question 1

Finding a trig ratio in terms of trig ratios of acute angles

Answer 1

Work out the smallest acute angle to the x-axis
Write what quadrant it’s in so what values are +/-
Apply that to the type of ratio you have

Question 2

Unit circle

Answer 2

A circle such that the radius is 1 and the centre is the origin
x-coordinate = cos θ
y-coordinate = sin θ
Where θ is the clockwise angle between the positive x-axis and the line

Question 3

Quadrant 1 of the unit circle

Answer 3

+ sin, cos and tan

Question 4

Unit circle quadrant 2

Answer 4

\+ sin, - cos and tan
Where θ is the anti-clockwise angle from the negative x-axis
sin(180-θ) = sin θ
cos(180-θ) = -cos θ
tan(180-θ) = -tan θ

Question 5

Unit circle quadrant 3

Answer 5

\+ tan, - cos and sin
Where θ is the clockwise angle from the negative x-axis
sin(180+θ) = -sin θ
cos(180+θ) = -cos θ
tan(180+θ) = tan θ

Question 6

Unit circle quadrant 4

Answer 6

\+ cos, - sin and tan
Where θ is the anti-clockwise angle from the positive x-axis
sin(360-θ) = -sin θ
cos(360-θ) = cos θ
tan(360-θ) = -tan θ

Question 7

Trig Identities

Answer 7

sin^2 x + cos^2 x = 1

tan x = sinx/cosx

Question 8

Using one trig value and the type of angle to find the others

Answer 8

Rearrange the identities for one

If it is squared, use the quadrant to see whether it’s positive or negative

Question 9

What must you manipulate for trig identity proofs

Answer 9

The left hand side to make it equal the RHS

Question 10

Solving sin/cos/tan θ = x in the interval 0 <= θ

Answer 10

1. Find the value given by your calculator
2. See which quadrants it is +/- in
3. Apply (180-θ), (180+θ) or (360-θ) depending on part 2

Question 11

What to do if your range is -180 <= θ <= 180

Answer 11

Go counter clockwise for quadrants 3/4

Question 12

When there is an expression involving θ inside the function

Answer 12

1. Apply the expression to the max and min values
2. Apply the inverse and find all outputs between the intervals
3. Manipulate so each value is in terms of θ

Question 13

Solving a quadratic involving sin/cos/tan

Answer 13

Let y = sin/cos/tan x
Factorise and solve for y
See what values or x give each sin/cos/tan x in the given range