Trigonometry Practice questions Part –I

1. If sin x + sin² x +sin³ x =1, then the value of  cos⁶ x −4cos⁴ x +8cos² x =

       (A) 0                     (B) 2                 (C)  4                (D) 8

First Method:

The given equation can be written as

Sin x (1+sin² x)= 1– sin² x =cos² x

⇒ sin x (2 – cos² x) = cos² x

⟹sin² x (2– cos² x)² = cos⁴ x  [after squaring both sides]

⟹(1– cos² x) (4– 4 cos² x + cos⁴ x)= cos⁴ x

⟹ 4– 4 cos² x + cos⁴ x – 4 cos² x + 4 cos⁴ x – cos⁶ x =  cos⁴ x

⟹ cos⁶ x −8cos⁴ x +4cos² x = 4

Answer is option C

Second Method:

If x= 30°

sin x + sin² x +sin³ x = ½ + ¼ +1/8  = 7/8 ∼ 1

Here it is less than 1 but in second expression , it will more than 4

Because when sine decreases from 90° to 0° then cosine will increases in same range. i.e. sin 30° =1/2 but cos 30° =√3/2

⟹ cos⁶ x −4cos⁴ x +8cos² x = (√3/2)⁶ – 8(√3/2)⁴ +4 (√3/2)²

⟹ 27/32 – 4× 9/16 +8× ¾ = 147/32 which is more than 4

So we can choose option  C

You can solve many questions by taking value. But in sine and cosine you should careful at 45°

More questions for practice:

1.  If cot α +tan α =m and 1/ cos α – cosα = n then

       (A)   m (mn²)^1/3 – n (nm²)^1/3 = 1

      (B)  m (m²n)^1/3 – n (mn²)^1/3 = 1

      (C) n (mn²)^1/3 – m (nm²)^1/3 = 1

     (D) n (mn²)^1/3 – m (mn²)^1/3 = 1

2. If sin x + sin² x = 1,  then the value of cos¹² x + 3cos¹⁰ x +3cos⁸ x + cos⁶ x –1

    (A) 0                (B) 1                (C) –1               (D) 2

3.   If a cos³ α + 3a cos α sin² α = m  and a sin³  α + 3a cos² α sin α = n then

      (m + n)^2/3 + (m– n )^1/3  is equal to

               (A) 2a²              (B) 2a^1/3               (C) 2a^2/3              (D) 2a³

4.   6(sin⁶ θ + cos⁶ θ) – 9 (sin⁴ θ + cos⁴ θ) is equal to

       (A)  –1                 (B)  1                     (C) –3                     (D)    3

5. (1 + tan α tan β )² + (tan α – tanβ )²  is equal to

        (A)  tan² α +  tan² β                           (B)   cos² α   cos² β

        (C)  sec² α    sec² β                            (D)  )  tan² α   tan² β

6.  If cos 1° cos 2° cos 3° cos 4° ………. cos 179° = x + 1, then x=

        (A)  –1              (B)  0             (C)   1                   (D)  none of these

7.   If cos x + sin x = √2 cos x then tan² x + 2 tan x =

         (A) 0                    (B)   1             (C)      2                   (D) 3

8.     If A = sin⁸ θ +  cos¹⁴ θ, then for all values of θ

          (A)  A > 1            (B)  A ≥ 1          (C) A < 1                   (D) A ≤ 1

9.     If θ is an acute angle and sin θ = cos θ , the value of  2tan² θ + sin² θ – 1 is

          (A) ½             (B)  3/2            (C)   0              (D)   1

10.   In an acute angled triangle ABC, if tan ( A + B – C ) = 1 and sec ( B + C – A)   = 2 , value of B =

           (A)   60°  (B) 52 (1/2 )°    (C)  67 (1/2)°          (D) 57  (1/2)°

11. An isosceles triangle ABC is right-angled at B. D is a point inside the triangle ABC. P  and Q are the feet of the perpendiculars drawn from D on the sides AB and AC respectively of Δ ABC. If AP = a cm, AQ = b cm and ∠BAD = 15º, sin75º =

           (A) 2b/ √a       (B)  a/2b         (C) √3 a/ 2b     (D) 2a/ √3 b