Trig

2. QUESTIONS
TRIGONOMETRY

3. Concepts
Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch
of mathematics that studies relationships between side lengths and angles of
triangles.
The trigonometric ratios of a triangle are also called trigonometric functions.
Sine, cosine, and tangent are 3 important trigonometric functions.
The six trigonometric functions are sine, cosine, secant, co-secant, tangent and
co-tangent.

4. Concepts
Formulas of Multiple Angles in Trigonometric Functions
sin(2x) = 2 sin x cos x
sin(3x) = 3 sin x - 4 sin3x = sin x ( - 1 + 4 cos2x )
sin(4x) = cos x ( 4 sin x - 8 sin3x ) = sin x ( - 4 cos x + 8 cos3x )
sin(5x) = 5 sin x -20 sin3x + 16 sin5x = sin x ( 1 - 12 cos2x + 16 cos4x )
cos(2x) = cos2x - sin2x = - 1 + 2 cos2x
cos(3x) = cos3x - 3 cos x sin2x = -3 cos x + 4 cos3x
cos(4x) = cos4x - 6 cos2x sin2x +sin4x = 1 - 8 cos2x + 8 cos4x

5. Concepts
Formulas of Multiple Angles in Trigonometric Functions
cos(5x) = cos5x - 10 cos3x sin2x + 5 cos x sin4x = 5 cos x - 20 cos3x + 16 cos5x
tan(2x) = 2 tan x / (1 - tan2x)
tan(3x) = ( 3 tan x - tan3x ) / (1 - 3 tan2x)
tan(4x) = ( 4 tan x - 4 tan3x ) / (1 - 6 tan2x + tan4x)

6. Concepts
Trigonometric ratios of right angled triangle ABC with ∠B = 90° :
sin A = sine of ∠A = (side opposite to ∠A) / hypotenuse = (BC/AC)
cos A = cosine of ∠A = (side adjacent to ∠A) / hypotenuse = (AB/AC)
tan A = tangent of ∠A = (side opposite to ∠A) / (side adjacent to ∠A)=(BC/AB)
cosec A = cosecant of ∠A = 1/ sin A
sec A = secant of ∠A = 1/ cos A
cot A = cotangent of ∠A = 1/ tan A

7. Question: 01
If tan4θ + tan2θ = 1, then the value of cos4θ + cos2θ is
A. 1/4
B. 1/2
C. 1
D. 1/3
Answer: C

8. Question: 02
The value of sin (45° + θ) – cos (45° – θ) is
A. 0
B. 1
C. 2cosθ
D. 2sinθ
Answer: A

9. Question: 03
If cot A + cosec A = and A is an acute angle, then the value of cos A is
A. 4/5
B. 2/3
C. 1/2
D. 1/4
Answer: A

10. Question: 04
Evaluate : ( Cot4 Θ – Cosec4 Θ + Cot2 Θ + Cosec2 Θ )
A. 1
B. 0
C. -1
D. -2
Answer: B

11. Question: 05
If cosΘ + secΘ = 2 ,then the value of cos68Θ + sec68Θ equal to
A. 1
B. 2
C. 4
D. 3
Answer: B

12. Question: 06
If 8 sin x = 4 + cos x, the values of sin x are :
A. 3/5 , 5/13
B. 3/5, 15/13
C. -2/5, 5/13
D. 2/5, 5/13
Answer: A

13. Question: 07
If tanΘ + cotΘ = 16, then find the ratio of tan2Θ + cot2Θ to tan2Θ + cot2Θ + 20
tanΘ.cotΘ
A. 64 : 65
B. 127 : 137
C. 107 : 137
D. 120 : 138
Answer: B

14. Question: 08
If sin 3A = cos (A – 26°), where 3A is an acute angle then the value of A is
A. 29 degree
B. 19 degree
C. 16 degree
D. 18 degree
Answer: A

15. Question: 09
The value of (sin 39°) / (cos 51°) + 2 tan11° tan31° tan45° tan59° tan79° – 3
(sin2 21° + sin2 69°) is :
A. 0
B. 1
C. -1
D. 1/2
Answer: A

16. Question: 10
If cos2 Θ / (cot2 Θ – cos2 Θ) = 3 and 0° < Θ < 90°, then the value of Θ is :
A. 60 degree
B. 45 degree
C. 30 degree
D. 90 degree
Answer: A

17. Question: 11
If A = tan 11° tan 29°, B = 2 cot 61° cot 79°, then :
A. A = 2B
B. 2A = B
C. 3A = 2B
D. -A = 2B
Answer: B

18. Question: 12
The value of cot 10° . cot 20° . cot 60° . cot 70° . cot 80° is
A. 1/√3
B. √3
C. 1
D. 0
Answer: A

19. Question: 13
The minimum value of 2 sin2 Θ + 3 cos2 Θ is
A. 0
B. 1
C. 2
D. -1
Answer: C

20. Question: 14
What is cosec (75° + Θ) – sec (15° – Θ) – tan (55° + Θ) + cot (35° – Θ) equal to?
A. 0
B. 1
C. -1
D. 1/2
Answer: A

21. Question: 15
If cos A + cos2 A = 1, then what is the value of 2(sin2 A + sin4 A)?
A. 3/2
B. 2
C. 0
D. -1
Answer: B