08 - Trigonometric Functions, Identities And Equations

St Joseph’s Institution Secondary Four Mathematics TOPIC − Trigonometric Functions, Identities and Equations Name:_____________________________________ ( Q1)

Q2)

) Class: ___________

Find all angles between 0° and 360° inclusive which satisfy the following equations 2 tan x = 3 sin x 2 cos 2 x + 3 cos x + 1 = 0 a) e) b)

2 cos 2 x + 3 sin x = 3

f)

sin( 2 x − 30 ) = cos 2 x

c)

7 sin 2 x + cos 2 x = 5 sin x

g)

sin 5 x − sin 3 x + sin x = 0

d)

sin 2 x + cos x = 0

h)

3 sin x + 4 cos x = 1

Prove the following identities a)

( sec x + tan x ) 2 ≡ 1 + sin x

b)

1 1 + ≡1 2 1 + tan x 1 + cot 2 x

c)

cot x +

d)

sin 4 x − cos 4 x ≡ sin 2 x − cos 2 x

e)

cos ec 2 x − 2 ≡ cos 2 x cos ec 2 x

f)

tan A + tan B sin( A + B ) ≡ tan A − tan B sin( A − B )

g)

2 cos ec 2 2 x − cos ec 2 x ≡ −2 cot 2 x cos ex 2 x

h)

(1 + 2 sin 2x )( cos x − sin x ) ≡ cos 3 x + sin 3 x

i)

sin 2 x + cos 2 x − 1 1 − tan x ≡ sin 2 x + cos 2 x + 1 1 + cot x

© Jason Ingham 2009

1 − sin x

sin x ≡ cos ec x 1 + cos x

1

Q3)

Q4)

Q5)

Simplify the expression ( tan θ − cot θ ) sinθ cos θ + 2 cos 2 θ . Given that 3 tanθ = 4 and that θ is acute, find, without using a calculator, sinθ + 3 cos θ the value of . 2 cos 2 θ − sinθ Express y = 3 cos 2 x − sin 2 x in the form R cos( x ± α ) where R > 0 and 0° < α < 90° . Hence, find i) the minimum value of y and the corresponding value of x for 0° ≤ x ≤ 360° ; ii)

the least value of

1 and the corresponding value of x for y

0° ≤ x ≤ 90° ; iii)

the range of values for y .

sin( A − B ) 3 = , prove that tan A + 5 tan B = 0 . Hence, solve the sin( A + B ) 2 equation 2 sin( A − 30°) = 3 sin( A + 30°) for 0° ≤ A ≤ 360° .

Q6)

Given that

Q7)

Given that cos 2 x = i)

1 , calculate without using a calculator, the values of 9 cos 4 x

ii)

tan 2 x

iii)

sin x .

Q8)

Find all values of A for which 1 + sin 2 A + cos 2 A = 0 for 0 ≤ A ≤ 11.

Q9)

Given that sin A = a and A is obtuse, express the following in terms of a cos 4 A i) ii)

π  tan − A  2 

iii)

cos ec ( 2π − A )

2

Q10) Solve the equation ( cos 4θ + cos θ ) + ( sin 4θ + sinθ ) = 2 3 sin 3θ for 0 ≤θ ≤ 3. 2

Q11) (a)

2

Prove the identity ( 2 + sin 2x ) cos x + (1 + cos 2 x ) sin x ≡ 2(1 + sin 2x ) cos x

(b)

Solve the equation 2 cos 2 θ + sin 2θ = 2 for 0° ≤ θ ≤ 360° .

(c)

Solve the equation cot θ + 4 tanθ = 4 cos ec θ for 0° ≤ θ ≤ 360° .

Q12) Given that 5 cos 2 A − 12 sin A cos A = A + B cos( 2 A + λ ) for all real values of A, find the value of A, of B and of λ for B > 0 and 0° ≤ λ ≤ 90° . Hence find the solution of the equation 5 cos 2 A − 12 sin A cos A = 2 for 0° ≤ A ≤ 360° . Q13) (a)

Find the values of x between 0° and 360° inclusive for which sin 2 x − 8 cos 2 x = 2 cos x .

(b)

Prove the identity sin 2 5θ − sin 2 3θ ≡ sin 8θ sin 2θ .

(c)

Given that sin( A + B ) =

Q14) (a)

4 5 and sin( A − B ) = where ( A + B ) and 5 13 ( A − B ) are angles between 0° and 90° , prove that tan 2A = 63 . 16

Prove that cot ( 45° − A ) =

cot A + 1 . Hence show that cot A + 1

cot 15° = 2 + 3 . (b)

(c)

4 4 Solve the equation sin B + cos B =

3 for 0° ≤ B ≤ 360° . 4

Express 3 cos θ + sinθ in the form R cos(θ − α ) where R > 0 and π 0 < α < . Hence or otherwise 2 i) show that 2 − 3 cos θ − sinθ ≥ 0 for all values of θ , ii)

solve the equation 2 − 3 cos θ − sinθ = 0 for 0° ≤ θ ≤ 360° .

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