Express cos θ + 3 sin θ in the form R cos ( θ - α ) , where R > 0 and 0 < α < π 2 Give the exact values of R and α .

Given that cos θ + 3 sin θ = R cos θ cos α + R sin θ sin α Equating coefficients of cos θ : R cos α = 1 . ......(i) Equating coefficients of sin θ : R sin α = 3 . .......(ii) Dividing equation (ii) with equation (i) ⇒ tan α = 3 ⇒ α = π 3 On squaring and adding equations (i) and (ii) we get R 2 1 = 1 2 + ( 3 ) 2 = 4 . ⇒ R = 2 ⇒ cos θ + 3 sin θ = 2 cos θ - π 3 Therefore, cos θ + 3 sin θ can be expressed as 2 cos θ - π 3 .

Express cos θ + 3 sin θ in the form R cos ( θ - α ) , where R > 0 and 0 < α < π 2 Give the exact values of R and α . Hence prove that 1 ( cos θ + 3 sin θ ) 2 ≡ 1 4 sec 2 θ - π 3 .

Given that cos θ + 3 sin θ = R cos θ cos α + R sin θ sin α Equating coefficients of cos θ : R cos α = 1 . ......(i) Equating coefficients of sin θ : R sin α = 3 . .......(ii) Dividing equation (ii) with equation (i) ⇒ tan α = 3 ⇒ α = π 3 ⇒ R 2 1 = 1 2 + ( 3 ) 2 = 4 . ⇒ R = 2 ⇒ cos θ + 3 sin θ = 2 cos θ - π 3 Therefore, cos θ + 3 sin θ can be expressed as 2 cos θ - π 3 . Now, cos θ + 3 sin θ = 2 cos θ - π 3 ⇒ 1 ( cos θ + 3 sin θ ) 2 = 1 2 2 cos 2 θ - π 3 = 1 4 sec 2 θ - π 3

EASY

AS and A Level

IMPORTANT

Earn 100

Express $4 \sin θ - 6 \cos θ$ in the form $R \sin (θ - α),$ where $R > 0$ and $0 ° < α < 90 °$ Give the exact value of $R$ and the value of $α$ correct to $2$ decimal places. Hence solve the equation $4 \sin θ - 6 \cos θ = 3$ for $0 ° < θ < 180 ° .$

Important Questions on Trigonometry

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 3 - Trigonometry>EXERCISE 3E>Q 4

Express $4 \sin θ - 6 \cos θ$ in the form $R \sin (θ - α),$ where $R > 0$ and $0 ° < α < 90 °$ Give the exact value of $R$ and the value of $α$ correct to $2$ decimal places. Find the greatest and least values of $(4 \sin θ - 6 \cos θ)^{2} - 3$ as $θ$ varies.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 3 - Trigonometry>EXERCISE 3E>Q 5

Express $3 \sin θ + 4 \cos θ$ in the form $R \sin (θ + α),$ where $R > 0$ and $0 ° < α < 90 ° .$ Give the value of $α$ correct to $2$ decimal places.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 3 - Trigonometry>EXERCISE 3E>Q 5

Express $3 \sin θ + 4 \cos θ$ in the form $R \sin (θ + α),$ where $R > 0$ and $0 ° < α < 90 ° .$ Give the value of $α$ correct to $2$ decimal places. Hence solve the equation $3 \sin θ + 4 \cos θ = 2$ for $0 ° < θ < 360 ° .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 3 - Trigonometry>EXERCISE 3E>Q 5

Find the least value of $3 \sin θ + 4 \cos θ + 3$ as $θ$ varies.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 3 - Trigonometry>EXERCISE 3E>Q 6

Express

\cos θ + \sqrt{3} \sin θ

in the form

R \cos (θ - α),

where

R > 0

and

0 < α < \frac{π}{2}

Give the exact values of

R

and

α .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 3 - Trigonometry>EXERCISE 3E>Q 6

Express

\cos θ + \sqrt{3} \sin θ

in the form

R \cos (θ - α),

where

R > 0

and

0 < α < \frac{π}{2}

Give the exact values of

R

and

α .

Hence prove that

\frac{1}{(\cos θ + \sqrt{3} \sin θ)^{2}} \equiv \frac{1}{4} \sec^{2} (θ - \frac{π}{3}) .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 3 - Trigonometry>EXERCISE 3E>Q 7

Express $8 \sin 2 θ + 4 \cos 2 θ$ in the form $R \sin (2 θ + α),$ where $R > 0$ and $0 ° < α < 90 ° .$ Give the exact value of $R$ and the value of $α$ correct to $2$ decimal places.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 3 - Trigonometry>EXERCISE 3E>Q 7

Express $8 \sin 2 θ + 4 \cos 2 θ$ in the form $R \sin (2 θ + α),$ where $R > 0$ and $0 ° < α < 90 ° .$ Give the exact value of $R$ and the value of $α$ correct to $2$ decimal places. Hence solve the equation $8 \sin 2 θ + 4 \cos 2 θ = 3$ for $0 ° < θ < 360 ° .$

Express 4sinθ-6cosθ in the form Rsin(θ-α), where R>0 and 0°<α<90° Give the exact value of R and the value of α correct to 2 decimal places. Hence solve the equation 4sinθ-6cosθ=3 for 0°<θ<180°.

Important Questions on Trigonometry

Learn and Prepare for any exam you want !

Express $4 \sin θ - 6 \cos θ$ in the form $R \sin (θ - α),$ where $R > 0$ and $0 ° < α < 90 °$ Give the exact value of $R$ and the value of $α$ correct to $2$ decimal places. Hence solve the equation $4 \sin θ - 6 \cos θ = 3$ for $0 ° < θ < 180 ° .$