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Chapter 7: Trigonometric Ratios of Compound Angles – Exercise 7.2

Trigonometric Ratios of Compound Angles – Exercise 7.2 – Q.1

Let f(θ) = 12 sin θ - 5 cos θ

We know that

Trigonometric Ratios of Compound Angles – Exercise 7.2 – Q.1

Hence, minimum and maximum values of 12 sin θ - 5 cos θ are -13 respectively.

Let f(θ) = 12 cos θ + 5 sin θ + 4

We know that

trigonometric-ratios-of-compound-angles–exercise-7.2–q.1(i)

⟹ -13 ≤ 12 cos θ + 5sin θ ≤ 13

⟹ -13 + 4 ≤ 12 cos θ + 5sinθ + 4 ≤ 13 + 4

⟹ -9 ≤ 12 cos θ + 5sinθ + 4 ≤ 17

⟹ -9 ≤ f(θ) ≤ 17

Hence, minimum and maximum values of 12 cos θ + 5 sin θ + 4 are – 9 and 17 respectively.

trigonometric-ratios-of-compound-angles–exercise-7.2–q.1(ii)

We know that

trigonometric-ratios-of-compound-angles–exercise-7.2–q.1(iii)

⟹ -3 ≤f (θ) ≤ 11

Let f(θ) = sin θ - cos θ + 1. Then,

f(θ) = sin θ + (-1) cos θ + 1

= (-1) cos θ + sin θ + 1

We know that

trigonometric-ratios-of-compound-angles–exercise-7.2–q.1(iv)

Hence, minimum and maximum values of sin θ - cos θ + 1 are 1 - √2  and 1 + √2 respectively.

 

Trigonometric Ratios of Compound Angles – Exercise 7.2 – Q.2

Let f(θ) = √3 sin θ - cos θ

Trigonometric Ratios of Compound Angles – Exercise 7.2 – Q.2

Again,

Let f(θ) = cos θ - sin θ

trigonometric-ratios-of-compound-angles–exercise-7.2–q.1(ii)

Again,

 trigonometric-ratios-of-compound-angles–exercise-7.2–q.1(iv)

Let f(θ) = 24 cos θ + 7 sin θ

trigonometric-ratios-of-compound-angles–exercise-7.2–q.1(v)

Again,

 trigonometric-ratios-of-compound-angles–exercise-7.2–q.1(vi)

 

Trigonometric Ratios of Compound Angles – Exercise 7.2 – Q.3

We have

sin100° - sin10°

Trigonometric Ratios of Compound Angles – Exercise 7.2 – Q.3

 

Trigonometric Ratios of Compound Angles – Exercise 7.2 – Q.4

Trigonometric Ratios of Compound Angles – Exercise 7.2 – Q.4

Dividing and multiplying the above equation with above value

 trigonometric-ratios-of-compound-angles–exercise-7.2–q.4(i)

We know that maximum and minimum value of any cosine term is +1 and -1

 trigonometric-ratios-of-compound-angles–exercise-7.2–q.4(ii)


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