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Finding the Unknown Angle

Problems on finding the unknown angle using trigonometric identities.

1. Solve: tan θ + cot θ = 2, where 0° < θ < 90°.

Solution:

Here, tan θ + cot θ = 2

⟹ tan θ + 1tanθ = 2

tan2θ+1tanθ = 2

⟹ tan2 θ + 1 = 2 tan θ

⟹ tan2 θ - 2 tan θ + 1 = 0

⟹ (tan θ - 1)2 = 0

⟹ tan θ – 1 = 0

⟹ tan θ = 1

⟹ tan θ = tan 45°

⟹ θ = 45°.

Therefore, θ = 45°.


2. Is sinθ1cosθ + sinθ1+cosθ = 4 an identity? If not, find θ (0° < θ < 90°).

Solution:

Here, LHS = sinθ(1+cosθ)+sinθ(1cosθ)(1cosθ)(1+cosθ)

                = 2sinθ1cos2θ

                = 2sinθsin2θ, [using trigonometric identities, sin2 θ + cos2 θ = 1]

                = 2sinθ

Thus, the given equality becomes 2sinθ = 4.

Now, if the equality holds true for all values of θ then the equality is an identity.

Let us take (arbitrarily) θ = 45°.

So, 2sin45° = 212 = 2√2

So, sin θ ≠ 4.

Therefore, the equality is not an identity.

It is an equation. Then, from the equation we have,

2sinθ = 4

⟹ sin θ = 12

⟹ sin θ = sin 30°

Therefore, θ = 30°.


3. If 5 cos θ + 12 sin θ = 13, find sin θ.

Solution:

5 cos θ + 12 sin θ = 13

⟹ 5 cos θ = 13 - 12 sin θ

⟹ (5 cos θ)2 = (13 – 12 sin θ)2

⟹ 25 cos2 θ = 169 - 312 sin θ + 144 sin θ2

⟹ 25(1 - sin2 θ) = 169 - 312 sin θ + 144 sin θ2, [using trigonometric identities, sin2 θ + cos2 θ = 1]

⟹ 25 – 25 sin2 θ = 169 – 312 sin θ + 144 sin θ2,

⟹ 169 sin2 θ – 312 sin θ + 144 = 0

⟹ (13 sin θ – 12)2 = 0

Therefore, 13 sin θ – 12 = 0

⟹ sin θ = 1213.

Finding the Unknown Angle

4. If 3sin θ - cos θ = 0, prove that tan 2θ = 2tanθ1tan2θ.

Solution:

Here, 3sin θ - cos θ = 0

⟹ sinθcosθ = 13

⟹ tan θ = 13

⟹ tan θ = tan 30°

⟹ θ = 30°

Therefore, tan 2θ = tan (2 × 30°) = tan 60° = √3

Now, 2tanθ1tan2θ = 2tan30°1tan230°

                   = 2×131(13)2

                   = 23113

                   = 2323

                   = 23 × 32

                   = √3.

Therefore, tan 2θ = 2tanθ1tan2θ. (proved)






10th Grade Math

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