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5.04 Trigonometric identities

Worksheet
Trigonometric identities
1

Express \tan x in terms of \cos x and \sin x .

2

Find the exact value of \tan 10 \degree - \dfrac{\sin 10 \degree}{\cos 10 \degree}.

3

Find the value of \cos x given the following:

a

\sin x = - \dfrac{\sqrt{3}}{2} and \tan x = -\sqrt{3}

b

\sin x = \dfrac{b}{c} and \tan x = \dfrac{b}{a}

4

Find the value of \sin x given the following:

a

\cos x = \dfrac{15}{17} and \tan x = \dfrac{8}{15}

b

\cos x = - \dfrac{5}{13} and \tan x \gt 0

5

Find the value of \tan x given the following:

a
\sin x = \dfrac{4}{5} and \cos x = \dfrac{3}{5}
b
\sin x = \dfrac{2}{3} and \cos x \lt 0
6

Simplify the following expressions:

a

\tan \theta \cos \theta

b

\sin \left(90 \degree - y\right) \tan y

c

\dfrac{\sin \theta - \cos \theta}{\cos \theta}

d

\dfrac{\tan \theta}{\text{cosec } \theta \sec \theta}

e
\sec \theta \cot \theta
f
\text{cosec }\theta \tan \theta
7

Find the exact value of \tan x given the following equations:

a
\sin x = 2 \cos x
b
17 \cos x - 31 \sin x = 0
c
6 \sin x - 11 \cos x = 0
d
2 \sin^2 x - \cos^2 x =0
Pythagorean identities
8

State whether the following statements are correct:

a

\sin ^{2}\theta + \cos ^{2}\theta = 1

b

\tan ^{2}x + 1 = \sec ^{2}x

c

\sin ^{2}\theta + \cos ^{2}\theta = 2

d

1 + \cot ^{2}\theta = \sec ^{2}\theta

9

Find the exact value of the following:

a
\sin ^{2}\left(20 \degree\right) + \cos ^{2}\left(20 \degree\right)
b
\tan ^{2}\left(40 \degree\right) - \sec ^{2}\left(40 \degree\right)
c
\text{cosec} ^{2}\left(50 \degree\right) - \cot ^{2}\left(50 \degree\right)
d
4\sin ^{2}\left(20 \degree\right) + 4\cos ^{2}\left(20 \degree\right)
10

Simplify the following expressions:

a

\left(\cos \theta - 1\right) \left(\cos \theta + 1\right)

b

\left(\cos \theta - \sin \theta\right)^{2}

c

\cos \theta \sin ^{2}\left(\theta\right) - \cos \theta

d

\left(3 - \cos x\right)^{2} + \sin ^{2}\left(x\right)

e

\left(1 + \tan ^{2}\left(u\right)\right) \left(1 - \sin ^{2}\left(u\right)\right)

f

\left(1 - \sec ^{2}\left(\theta\right)\right) \cot \theta

g

\text{cosec } ^{2}\left(\theta\right) - \cot ^{2}\left(\theta\right)

h

\cos \theta \left(\sec \theta - \cos \theta\right)

i

\left(1 - \text{cosec } \theta\right) \left(1 + \sin \theta\right)

j

\sec ^{2}\left(x\right) \left(\cos ^{2}\left(x\right) - 1\right)

k

\dfrac{1}{1 - \cos \theta} + \dfrac{1}{1 + \cos \theta}

l

\dfrac{\sin ^{2}\left(\theta\right)}{1 - \sin ^{2}\left(\theta\right)}

m

\dfrac{1}{1 - \sin x} \times \dfrac{1}{1 + \sin x}

n

\dfrac{1}{1 - \cos ^{2}\left(x\right)} - 1

o

\dfrac{1 - \sin ^{2}\left(\theta\right)}{\sin ^{2}\left(\theta\right) + \cos ^{2}\left(\theta\right)}

p

\dfrac{\sin \theta}{1 + \cos \theta} + \dfrac{1 + \cos \theta}{\sin \theta}

q

\dfrac{1}{1 + \tan ^{2}\left(x\right)}

r

\tan \theta + \dfrac{1}{\tan \theta}

s

\dfrac{\sec ^{2}\left(x\right) - 1}{\text{cosec } ^{2}\left(x\right) - 1}

t

\dfrac{1}{\text{cosec } ^{2}\left(\theta\right)} + \dfrac{\cos \theta}{\sec \theta}

u

\dfrac{1}{\text{cosec } ^{2}\left(\theta\right) - 1}

v

\dfrac{1}{\cos ^{2}\left(x\right)} - \dfrac{1}{\cot ^{2}\left(x\right)}

w

\left(\sec \theta - \text{cosec } \theta\right) \left(\cos \theta + \sin \theta\right)

11

Simplify \sqrt{a^{2} + x^{2}}, where x = a \tan \theta, a is a constant, and 0 \degree \lt \theta \lt 90 \degree.

12

If x = 4 \sin \theta and y = 3 \cos \theta, form an equation relating x and y that does not involve \sin \theta or \cos \theta.

13

State the values of x that are not in the domain of the identify 1 + \cot ^{2}\left(x\right) = \text{cosec } ^{2}\left(x\right)

14

If \sin \theta = x, express \dfrac{1 - \cos ^{2}\left(\theta\right)}{\sec ^{2}\left(\theta\right)} in terms of x.

Proofs
15

Prove the following identities:

a

\dfrac{\cos x \tan x}{\sin x} = 1

b

\dfrac{\sec \theta}{\tan \theta} = \text{cosec } \theta

c

\dfrac{1 - \sin ^{2}\left(x\right)}{\cos x} = \cos x

d

\dfrac{\sin \theta}{1 - \cos \theta} = \dfrac{1 + \cos \theta}{\sin \theta}

e

\dfrac{1 - \cot x}{1 + \cot x} = \dfrac{\tan x - 1}{\tan x + 1}

f

\left(\sin x + \cos x\right)^{2} = 1 + 2 \sin x \cos x

g

\sin A \cos A \tan A = \sin ^{2}\left(A\right)

h

\cos ^{4}\left(x\right) - \sin ^{4}\left(x\right) = 2 \cos ^{2}\left(x\right) - 1

i

5 \cos ^{2}\left(\theta\right) - 3 = 2 - 5 \sin ^{2}\left(\theta\right)

j

\dfrac{\left(1 + \sin \theta\right)^{2} + \cos ^{2}\left(\theta\right)}{1 + \sin \theta} = 2

k

\dfrac{\sin ^{2}\left(x\right) + \sin x \cos x}{\cos ^{2}\left(x\right) + \sin x \cos x} = \tan x

l

\dfrac{\sin x \cos \left(90 \degree - x\right)}{\cos x \sin \left(90 \degree - x\right)} = \tan ^{2}\left(x\right)

m

\tan \theta \sin \theta + \cos \theta = \sec \theta

n

\dfrac{\tan ^{2}\left(x\right) - 1}{\tan ^{2}\left(x\right) + 1} = 1 - 2 \cos ^{2}\left(x\right)

o

\dfrac{1 + \cot ^{2}\left(x\right)}{\text{cosec } x} = \text{cosec } x

p

\dfrac{\cos ^{2}\left(x\right)}{\sin x} = \text{cosec } x - \sin x

q

\tan x + \cot x = \text{cosec } x \sec x

r

\dfrac{\text{cosec } ^{4}\left(x\right) - \cot ^{4}\left(x\right)}{\text{cosec } ^{2}\left(x\right) + \cot ^{2}\left(x\right)} = 1

s

\dfrac{4 + \tan ^{2}\left(x\right) - \sec ^{2}\left(x\right)}{\text{cosec } ^{2}\left(x\right)} = 3 \sin ^{2}\left(x\right)

t

\dfrac{\cos \alpha}{1 + \sin \alpha} = \sec \alpha \left(1 - \sin \alpha\right)

u

\left(\sec x - \tan x\right)^{2} = \dfrac{1 - \sin x}{1 + \sin x}

v

\dfrac{\sin x}{\text{cosec } x - 1} = \dfrac{1 + \sin x}{\cot x}

16

Prove the following identities:

a

\tan ^{2}\left(y\right) - \sin ^{2}\left(y\right) = \tan ^{2}\left(y\right) \sin ^{2}\left(y\right)

b

\sin ^{2}\left(a\right) - \sin ^{2}\left(b\right) + \cos ^{2}\left(a\right) \sin ^{2}\left(b\right) - \sin ^{2}\left(a\right) \cos ^{2}\left(b\right) = 0

c

\left(1 - \sin ^{2}\left(x\right)\right) \left(1 + \sin ^{2}\left(x\right)\right) = 2 \cos ^{2}\left(x\right) - \cos ^{4}\left(x\right)

d

\sin \theta \left(1 + \tan \theta\right) + \cos \theta \left(1 + \cot \theta\right) = \dfrac{\sin \theta + \cos \theta}{\sin \theta \cos \theta}

17

A searchlight at the grand opening of a new car dealership casts a spot of light on a wall located 75 meters from the searchlight. The acceleration a of the spot of light is found to be a = 1200 \sec \theta \left( 2 \sec ^{2}\left(\theta\right) - 1\right). Show that this is equivalent to a = 1200 \left(\dfrac{1 + \sin ^{2}\left(\theta\right)}{\cos ^{3}\left(\theta\right)}\right).

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Outcomes

MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

MA11-4

uses the concepts and techniques of periodic functions in the solutions of trigonometric equations or proof of trigonometric identities

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