# Mathematics Questions and Answers – Trigonometric Functions of Sum and Difference of Two Angles-1

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Trigonometric Functions of Sum and Difference of Two Angles-1”.

1. cos(75°) =__________________
a) (1 – $$\sqrt{3}$$)/2$$\sqrt{2}$$
b) ($$\sqrt{3}$$ + 1)/2$$\sqrt{2}$$
c) ($$\sqrt{3}$$ – 1)/2$$\sqrt{2}$$
d) (-$$\sqrt{3}$$ – 1)/2$$\sqrt{2}$$

Explanation: cos(75°) = cos (45°+30°) = cos45° cos30° – sin45° sin30°
= (1/$$\sqrt{2}$$ * $$\sqrt{3}$$/2) – (1/$$\sqrt{2}$$ * 1/2) {cos(x + y)=cos x cos y – sin x sin y}
= ($$\sqrt{3}$$ – 1)/2$$\sqrt{2}$$.

2. cos(15°) =_____________
a) (1 – $$\sqrt{3}$$)/2$$\sqrt{2}$$
b) ($$\sqrt{3}$$ + 1)/2$$\sqrt{2}$$
c) ($$\sqrt{3}$$ – 1)/2$$\sqrt{2}$$
d) (-$$\sqrt{3}$$ – 1)/2$$\sqrt{2}$$

Explanation: cos(15°) = cos (45°-30°) = cos45° cos30° + sin45° sin30°
= (1/$$\sqrt{2}$$ * $$\sqrt{3}$$/2) + (1/$$\sqrt{2}$$ * 1/2) {cos(x – y)=cos x cos y + sin x sin y}
= ($$\sqrt{3}$$ +1)/2$$\sqrt{2}$$.

3. sin (75°) =__________________
a) (1 – $$\sqrt{3}$$)/2$$\sqrt{2}$$
b) ($$\sqrt{3}$$ + 1)/2$$\sqrt{2}$$
c) ($$\sqrt{3}$$ – 1)/2$$\sqrt{2}$$
d) (- $$\sqrt{3}$$ – 1)/2$$\sqrt{2}$$

Explanation: sin (75°) = sin (45°+30°) = sin45° cos30° + cos45° sin30°
= (1/$$\sqrt{2}$$ * $$\sqrt{3}$$/2) + (1/$$\sqrt{2}$$ * 1/2) {sin(x + y)=sin x cos y + cos x sin y}
= ($$\sqrt{3}$$ + 1)/2$$\sqrt{2}$$.
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4. sin(15°) =_________________
a) (1 – $$\sqrt{3}$$)/2$$\sqrt{2}$$
b) ($$\sqrt{3}$$ + 1)/2$$\sqrt{2}$$
c) ($$\sqrt{3}$$ – 1)/2$$\sqrt{2}$$
d) (- $$\sqrt{3}$$ – 1)/2$$\sqrt{2}$$

Explanation: sin (15°) = sin (45°-30°) = sin45° cos30° – cos45° sin30°
= (1/$$\sqrt{2}$$ * $$\sqrt{3}$$/2) – (1/$$\sqrt{2}$$ * 1/2) {sin(x – y)=sin x cos y – cos x sin y}
= ($$\sqrt{3}$$ -1)/2$$\sqrt{2}$$.

5. Is cos (90° – x) = sin x.
a) True
b) False

Explanation: cos (90° – x) = cos 90° cos x + sin 90° sin x {cos(x – y)=cos x cos y + sin x sin y}
= 0*cos x + 1*sin x
= sin x.

6. Is sin (90°+x) = cos x.
a) True
b) False

Explanation: sin (90°+x) = sin 90° cos x + cos 90° sin x {sin(x + y)=sin x cos y + cos x sin y}
= 1*cos x + 0*sin x
= cos x.

7. tan(75°) =___________________
a) 2+$$\sqrt{3}$$
b) 2-$$\sqrt{3}$$
c) 1+$$\sqrt{3}$$
d) $$\sqrt{3}$$-1

Explanation: tan (x +y) = (tan x + tan y)/(1- tan x tan y)
tan (45°+30°) = (tan 45° + tan 30°)/(1- tan 45° tan 30°)
tan 75° = (1+ 1/$$\sqrt{3}$$)/(1-1/$$\sqrt{3}$$) = ($$\sqrt{3}$$ + 1)/ ($$\sqrt{3}$$ – 1) = 2+$$\sqrt{3}$$.

8. tan(15°) =___________________
a) 2 + $$\sqrt{3}$$
b) 2 – $$\sqrt{3}$$
c) 1 + $$\sqrt{3}$$
d) $$\sqrt{3}$$ – 1

Explanation: We know, tan (x -y) = (tan x – tan y)/(1+ tan x tan y)
tan (45°-30°) = (tan 45° – tan 30°)/(1+ tan 45° tan 30°)
tan 75° = (1- 1/$$\sqrt{3}$$)/ (1+ 1/$$\sqrt{3}$$) = ($$\sqrt{3}$$ – 1)/ ($$\sqrt{3}$$ + 1) = 2-$$\sqrt{3}$$.

9. cot 75° =___________________________
a) 2+$$\sqrt{3}$$
b) 2-$$\sqrt{3}$$
c) 1+$$\sqrt{3}$$
d) $$\sqrt{3}$$-1

Explanation: We know, cot (x +y) = (cot x cot y -1)/(cot y + cot x)
cot(45°+30°) = (cot 45° cot 30°-1)/(cot 45° + cot 30°)
cot 75° = ($$\sqrt{3}$$ – 1)/($$\sqrt{3}$$ + 1) = 2-$$\sqrt{3}$$.

10. cot 15° =______________
a) 2+$$\sqrt{3}$$
b) 2-$$\sqrt{3}$$
c) 1+$$\sqrt{3}$$
d) $$\sqrt{3}$$-1

Explanation: We know, cot (x – y) = (cot x cot y +1)/cot y – cot x)
cot(45°-30°) = (cot 45° cot 30°+1)/(cot 45° – cot 30°)
cot 15° = ($$\sqrt{3}$$ + 1)/($$\sqrt{3}$$ – 1) = 2+$$\sqrt{3}$$.

11. Find cos 2x if sin x=1/2.
a) 1/2
b) 1/$$\sqrt{2}$$
c) $$\sqrt{3}$$/2
d) 1

Explanation: We know, cos 2x = cos2x – sin2x = 1-2sin2x {cos2x = 1-sin2x}
= 1-2(1/2)2 = 1-2(1/4) = 1-1/2 = 1/2.

12. Find cos 2x if cos x = 1/$$\sqrt{2}$$.
a) 1/2
b) 0
c) $$\sqrt{3}$$/2
d) 1

Explanation: We know, cos 2x = cos2x – sin2x = 2cos2x – 1 {sin2x = 1-cos2x}
= 2(1/$$\sqrt{2}$$)2-1 = 2(1/2) – 1 = 1-1 = 0.

13. Find cos 2x if tan x=1/$$\sqrt{3}$$.
a) 1/2
b) 0
c) $$\sqrt{3}$$/2
d) 1

Explanation: We know, cos 2x = cos2x – sin2x = (cos2x – sin2x)/(cos2x+sin2x) {1 = sin2x + cos2x}
= (1-tan2x)/(1+tan2x)
= (1-(1/$$\sqrt{3}$$)2)/(1+(1/$$\sqrt{3}$$)2)
= (1-1/3)/(1+1/3) = (2/3)/(4/3) = 1/2.

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