Cbse Class 10th Study Guides

UNIT-6 TRIGONOMETRY
"The mathematician is fascinated with the marvelous beauty of the forms he constructs, and in their beauty he finds everlasting truth."
If xcosθ – ysinθ = a, xsinθ + ycos θ = b, prove that x2+y2=a2+b2.

1.

Ans: xcosθ - y sinθ = a xsinθ + y cosθ = b Squaring and adding x2+y2=a2+b2.
2. Prove that sec2θ+cosec2θ can never be less than 2.

Ans: S.T Sec2θ + Cosec2θ can never be less than 2. If possible let it be less than 2. 1 + Tan2θ + 1 + Cot2θ < 2. ⇒ 2 + Tan2θ + Cot2θ ⇒ (Tanθ + Cotθ)2 < 2. Which is not possible.
3. If sinϕ = , show that 3cosϕ-4cos3ϕ = 0.

Ans: Sin ϕ = ½ ⇒ ϕ = 30o Substituting in place of ϕ =30o. We get 0.
4. If 7sin2ϕ+3cos2ϕ = 4, show that tanϕ = 1 .

Ans: If 7 Sin2ϕ + 3 Cos2ϕ = 4 S.T. Tanϕ
2 2 2

3 7 Sin ϕ + 3 Cos ϕ = 4 (Sin ϕ + Cos ϕ)
2

⇒ 3 Sin2ϕ = Cos2ϕ ⇒ Sin 2ϕ 1 = Cos 2ϕ 3

43

⇒ Tan2ϕ =
1 3

1 3

Tanϕ =

5.

If cosϕ+sinϕ =

cosϕ, prove that cosϕ - sinϕ =

sin ϕ.

Ans: Cosϕ + Sinϕ = 2 Cosϕ ⇒ ( Cosϕ + Sinϕ)2 = 2Cos2ϕ ⇒ Cos2ϕ + Sin2ϕ+2Cosϕ Sinϕ = 2Cos2ϕ ⇒ Cos2ϕ - 2Cosϕ Sinϕ+ Sin2ϕ = 2Sin2ϕ ⇒ (Cosϕ - Sinϕ)2 = 2Sin2ϕ Cos2ϕ or Cosϕ - Sinϕ = 2 Sinϕ.
6.

∴2Sin2ϕ = 2 - 2Cos2ϕ 1- Cos2ϕ = Sin2ϕ & 1 - Sin2ϕ =

If tanA+sinA=m and tanA-sinA=n, show that m2-n2 = 4

Ans: TanA + SinA = m TanA – SinA = n. 2 2 m -n =4 mn . m2-n2= (TanA + SinA)2-(TanA - SinA)2 = 4 TanA SinA RHS 4 mn = 4 (TanA + SinA)(TanA − SinA)
=4 =4 =4

Tan 2 A − Sin 2 A
Sin 2 A − Sin 2 ACos 2 A Cos 2 A

Sin 4 A Cos 2 A Sin 2 A = 4 TanA SinA =4 Cos 2 A ∴m2 – n2 = 4 mn

7.

If secA=

, prove that secA+tanA=2x or

.

44

Ans: Secϕ = x +

1 4x (Sec2ϕ= 1 + Tan2ϕ)

⇒ Sec2ϕ =( x +

1 2 ) 4x 1 2 Tan2ϕ = ( x + ) -1 4x 1 2 ) Tan2ϕ = ( x 4x Tanϕ = + x 1 4x

Secϕ + Tanϕ = x + = 2x or 8. 1 2x

1 1 + x4x 4x

If A, B are acute angles and sinA= cosB, then find the value of A+B.

Ans: A + B = 90o
a)Solve for ϕ, if tan5ϕ = 1.

9.

Ans: Tan 5ϕ = 1

⇒ϕ=

45 ⇒ ϕ=9o. 5

b)Solve for ϕ if

Sinϕ 1 + Cosϕ