Find the Values of sin18, cos18

The value of sin18° is equal to (√5-1)/4 and the value of cos18° is equal to √(10+2√5)/2. Here we will learn to find the values of sin18, cos18.

The sin 18 degree formula is given by

sin18=512

The cos 18 degree formula is given by

cos18=10+252

Value of sin18

Value of sin18

Question: Find the value of sin18°.

Answer:

Let us assume

θ = 18°

[We need to find sinθ]

⇒ 5θ = 90°

⇒ 3θ+2θ = 90°

⇒ 2θ = 90° -3θ

Taking sin on both sides, we get that

sin2θ = sin(90° -3θ)

⇒ sin2θ = cos3θ, by the formula sin(90-x)=cosx.

⇒ 2sinθ cosθ = 4cos3θ -3cosθ, using the formula sin2A= 2sinA cosA, and cos3A= 4cos3A -3cosA.

⇒ 2 sinθ cosθ – 4cos3θ + 3cosθ = 0

⇒ cosθ (2sinθ – 4cos2θ + 3) = 0

As cosθ = cos18° ≠ 0, we get that

2sinθ – 4cos2θ + 3 =0

⇒ 2sinθ – 4(1-sin2θ) + 3 =0 as sin2θ+cos2θ =1.

⇒ 4sin2θ + 2sinθ – 1 =0

So sin18° is a positive root of the quadratic equation 4x2+2x-1=0.

By the rule of quadratic equations,

sinθ = 2±4+162×4 = 1±54

As sinθ = sin18° lies in the first quadrant, its value is positive. In other words, we conclude that

sinθ = 1+54.

Thus, the value of sin18° is equal to (√5-1)/4.

Also Read: Values of sin15, cos 15, tan15

Values of sin75, cos75, tan75

Value of cos18

Value of cos18

Question: Find the value of cos18°.

Answer:

Let θ = 18°

Using the identity sin2θ + cos2θ =1, we have that

cos18° = cosθ = 1sin2θ

⇒ cos18° = 1sin218

⇒ cos18° = 1(512)2 as sin18° = (√5-1)/4 by above.

= 165+2514

= 10+254

So the value of cos18° is equal to 10+254.

Also Read: Sin3x in terms of sinx

FAQs

Q1: What is the value of sin 18 degrees?

Answer: The value of sin 18 degrees is equal to (√5-1)/4.

Q2: What is the value of cos 18 degrees?

Answer: The value of cos 18 degrees is equal to √(10+2√5)/2.

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