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
The cos 18 degree formula is given by
Value of sin18
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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 =
As sinθ = sin18° lies in the first quadrant, its value is positive. In other words, we conclude that
sinθ =
Thus, the value of sin18° is equal to (√5-1)/4.
Also Read: Values of sin15, cos 15, tan15
Value of cos18
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Question: Find the value of cos18°.
Answer:
Let θ = 18°
Using the identity sin2θ + cos2θ =1, we have that
cos18° = cosθ =
⇒ cos18° =
⇒ cos18° =
=
=
So the value of cos18° is equal to
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.