Trigonometry

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SSC 2015 paper questions

Q11) If 0° < θ < 90° then the value of sin θ ⁡ + cos ⁡θ is

1) equal to 1
2) greater than 1
3) less than 1
4) equal to 2

Let y = sin ⁡ θ + cos ⁡θ
y² = sin²⁡ θ + cos²⁡θ + 2 sinθ.cos⁡θ
= 1 + 2sin⁡θcos⁡θ
For 0° < θ < 90°,sin⁡θ < 1,cos⁡θ < 1
2 sin ⁡θ cos ⁡θ < 1
y² = 2
y < √2
Hence, option 3.

Q12) Maximum value of sin⁸θ + cos¹⁴θ for all real values of θ is

1) 1
2) 0
3) 1.414
4) 0.707

sin⁸θ + cos¹⁴θ = sin⁸θ + (1 - sin²θ)⁷
Maximum value of sin²θ = 1
So, maximum value of sin⁸θ + (1 - sin²θ)⁷ = 1 + 0 = 1
Hence, option 1.

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Trigonometric Identities
Q13) If then the value of is

1) -1
2) 0
3) 1
4) 2

for all integral values of x.

Hence, option 2.

Q14) In a right angled triangle XYZ right angled at Y, if then sec x + tan x is

Hence, option 2.

Q15) If cos ⁡ x + sec ⁡x = 2 then what is the value of cos⁽ⁿ⁺¹⁾⁡ x + sec⁽ⁿ⁺¹⁾⁡x where n is a positive integer?

1) 2⁽ⁿ⁺¹⁾
2) 2^(n-1)
3) 2ⁿ
4) 2

If then
So, cos⁽ⁿ⁺¹⁾⁡ x + sec⁽ⁿ⁺¹⁾⁡x = 2
Hence, option 4.

Q16) Consider the following statements
A) tan ⁡θ increases faster than sin ⁡θ as θ increases.
B) The value of sin θ ⁡ + cos ⁡θ is always greater than 1.
Which of the statements given above is/are correct?

1) Only A
2) Only B
3) Both A and B
4) Neither A nor B

Statement A is correct as tan θ ⁡ increases faster than sin ⁡θ as θ increases.
Statement B is not correct as sin θ ⁡ + cos ⁡θ can even be equal to 1.
Hence, option 1.

Q17) If where 0° < θ < 90° then tan θ is equal to

Hence, option 3.

Q18) If cos ⁡x + cos²x = 1 then the numerical value of sin¹²x + 3sin¹⁰x + 3sin⁸x + sin⁶x - 1 is

1) 1
2) 0
3) -1
4) 2

cos ⁡ x + cos²x = 1
cos ⁡ x = 1 - cos²x = sin²x
On cubing both sides,
cos ⁡x + cos²x = 1
(cos ⁡x + cos²x)³ = 1
cos⁶x + 3cos⁵x + 3cos⁴x + cos³x = 1
Put cos ⁡x = sin²x
sin¹²x + 3sin¹⁰x + 3sin⁸x + sin⁶x = 1
sin¹²x + 3sin¹⁰x + 3sin⁸x + sin⁶x - 1 = 0
Hence, option 2.

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