Question

If cos A = 21 29 , calculate sin A

Answer 1

Solution:

Given That:

$\rm \longrightarrow \cos \alpha = \dfrac{21}{29}$

We know that:

$\rm \longrightarrow \sin^{2} \alpha + \cos^{2} \alpha = 1$

Therefore:

$\rm \longrightarrow \sin^{2} \alpha = 1 - \cos^{2} \alpha$

$\rm \longrightarrow \sin\alpha = \sqrt{ 1 - \cos^{2} \alpha}$

Substituting the value of cos α, we get:

$\rm \longrightarrow \sin\alpha = \sqrt{ 1 - \dfrac{ {21}^{2} }{ {29}^{2} } }$

$\rm \longrightarrow \sin\alpha = \sqrt{ \dfrac{ {29}^{2} - {21}^{2} }{ {29}^{2} } }$

$\rm \longrightarrow \sin\alpha = \sqrt{ \dfrac{841- 441}{ {29}^{2} } }$

$\rm \longrightarrow \sin\alpha = \sqrt{ \dfrac{400}{ {29}^{2} } }$

$\rm \longrightarrow \sin\alpha = \sqrt{ \dfrac {{20}^{2} }{ {29}^{2} } }$

$\rm \longrightarrow \sin\alpha = \dfrac {20}{29}$

★ 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
• cosec θ = 1/sin θ
• sec θ = 1/cos 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