Question

if cos theta = root3 divide2, find sin theta and tan theta and also prove that sin squre theta +cos squre theta = 1​

Answer 1

Step-by-step explanation:

Given:

We are given that, cos theta = √3/2.

To do:

We need to do two things:

1. We need to find sin theta
2. We need to find tan theta
3. We need to prove that sin²theta+cos²theta = 1.

Solution:

(Please refer diagram for better understanding.)

→ cos theta = √3/2

We know that,

→ Base/Hypotenuse = cos theta

→ Base/Hypotenuse = √3/2

Let us assume:

• Base = √3 k
• Hypotenuse = 2k

Pythagoras theorem states that in a right angled triangle, the square of Hypotenuse is equal to the sum of square of other two sides.

→ (Hypotenuse)² = (Perpendicular)²+(Base)²

→ (Perpendicular)² = (Hypotenuse)²-(Base)²

→ (Perpendicular)² = (2k)²-(√3k)²

→ (Perpendicular)² = 4k²-3k²

→ (Perpendicular)² = 1k²

→ Perpendicular = √1k²

→ Perpendicular = 1k

Now, we know that:

→ sin theta = Perpendicular/Hypotenuse

→ sin theta = 1k/2k

→ sin theta = 1/2

We know that, tan theta is equal to sin theta by cos theta:

→ tan theta = sin theta/cos theta

→ tan theta = (1k/2k)/(√3 k/2k)

→ tan theta = 1k/√3 k

→ tan theta = 1/√3

Finally, let us prove our question:

→ sin²theta+cos²theta = 1

→ (1/2)²+(√3/2)²= 1

→ 1/4+3/4 = 1

→ 4/4 = 1

→ 1 = 1

Hence, proved!