Trigonometry studies relationships between side lengths and angles of triangles. The document defines trigonometric ratios and functions such as sine, cosine, and tangent. It provides formulas for multiple angles of trigonometric functions. It also defines trigonometric ratios for a right angled triangle and provides example problems testing knowledge of trigonometric concepts and formulas.

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