Question

1. Evaluate sin 18° (1) (ii )cos 72.​

Answer 1

Your answer is in attachment

Answer 2

Answer:

How to Find Sin 18 Value

We know that,

1 rad = (π/180) degrees

That means,

1 rad = 57.2958 degrees

18 rad = 1031.32 degrees

Thus, sin 18 rad = sin 1031.32°

Using trigonometric functions calculator, we can get the value of sin 1031.32°.

Hence, sin 18 rad = 0.769214…

We know that 1031.32° lies in the fourth quadrant, where sin is negative.

Therefore, sin 18 rad = -0.769214…

Also, we can represent the value of sin 18 in fraction in the same way as we did in the case of sin 18 degrees.

Cos 18 Value

Now, let’s learn how to find the value of cos 18 degrees.

Consider the identity sin2A + cos2A = 1,

Let us take A = 18°.

sin2(18°) + cos2(18°) = 1

cos2(18°) = 1 – sin2(18°)

= 1 – [(√5 – 1)/4]2

= 1 – [(1 + 5 – 2√5)/16]

= [16 – 6 + 2√5]/16

= (10 + 2√5)/16

Therefore,

cos18∘=10+25√16−−−−−−√=10+25√√4

Using the sin of 18 degrees value, we can write the other sin values as given in the below table

Exact Value of cos 72°

 

We will learn to find the exact value of cos 72 degrees using the formula of submultiple angles.

How to find the exact value of cos 72°?

Let, A = 18°                        

Therefore, 5A = 90° 

⇒ 2A + 3A = 90˚

⇒ 2A = 90˚ - 3A

Taking sine on both sides, we get 

sin 2A = sin (90˚ - 3A) = cos 3A 

⇒ 2 sin A cos A = 4 cos33 A - 3 cos A

⇒ 2 sin A cos A - 4 cos33 A + 3 cos A = 0

⇒ cos A (2 sin A - 4 cos22 A + 3) = 0 

Dividing both sides by cos A = cos 18˚ ≠ 0, we get

⇒ 2 sin A - 4 (1 - sin22 A) + 3 = 0

⇒ 4 sin22 A + 2 sin A - 1 = 0, which is a quadratic in sin A

Therefore, sin A = −2±−4(4)(−1)√2(4)−2±−4(4)(−1)2(4)

⇒ sin A = −2±4+16√8−2±4+168

⇒ sin A = −2±25√8−2±258

⇒ sin A = −1±5√4−1±54

sin 18° is positive, as 18° lies in first quadrant.

Therefore, sin 18° = sin A = √5−14√5−14

Now, cos 72° = cos (90° - 18°) = sin 18° = √5−14