Question

If 13 sinθ = 5 , find the values of tanθ and cosθ.
pls answer this​

Answer 1

Solution:

Given That:-

$\rm\longrightarrow 13\sin\theta=5$

$\rm\longrightarrow \sin\theta=\dfrac{5}{13}$

$\rm\longrightarrow \dfrac{Height}{Hypotenuse}=\dfrac{5}{13}$

Let us assume that:-

$\rm\longrightarrow Height=5k$

$\rm\longrightarrow Hypotenuse=12k$

Where k is a real number, k ≠ 0.

We know that:-

$\rm\longrightarrow Hypotenuse^{2}=Height^{2}+Base^{2}$

$\rm\longrightarrow Base^{2}=Hypotenuse^{2}-Height^{2}$

$\rm\longrightarrow Base^{2}=(13k)^{2}-(5k)^{2}$

$\rm\longrightarrow Base^{2}=169k^{2}-25k^{2}$

$\rm\longrightarrow Base^{2}=144k^{2}$

$\rm\longrightarrow Base=12k$

Therefore:-

$\rm\longrightarrow \cos\theta=\dfrac{Base}{Hypotenuse}=\dfrac{12k}{13k}=\dfrac{12}{13}$

$\rm\longrightarrow \tan\theta=\dfrac{Height}{Base}=\dfrac{5k}{12k}=\dfrac{5}{12}$

Which is our required answer.

Additional Information:

1. Relationship between sides and T-Ratios.

• sin θ = Height/Hypotenuse
• cos θ = Base/Hypotenuse
• tan θ = Height/Base
• cot θ = Base/Height
• sec θ = Hypotenuse/Base
• cosec θ = Hypotenuse/Height

2. Square formulae.

• sin²θ + cos²θ = 1
• cosec²θ - cot²θ = 1
• sec²θ - tan²θ = 1

3. Reciprocal Relationship.

• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ

4. Cofunction identities.

• sin(90° - θ) = cos θ
• cos(90° - θ) = sin θ
• cosec(90° - θ) = sec θ
• sec(90° - θ) = cosec θ
• tan(90° - θ) = cot θ
• cot(90° - θ) = tan θ

5. Even odd identities.

• sin -θ = -sin θ
• cos -θ = cos θ
• tan -θ = -tan θ