How do you find the exact value of cos 36° using the sum and difference, double angle, or half-angle formulas?
Double angle formulas are used to express the trigonometric ratios of double angles (2θ) in terms of trigonometric ratios of single angle (θ) and similarly, half-angle formulas are used to express the trigonometric ratios of half angles ((1/2)θ) in terms of trigonometric ratios of single angle (θ).
Answer: The exact value of cos 36° using the sum and difference, double angle, or half-angle formulas is cos 36° = (√5 +1) / 4
Let's look into the steps below.
Explanation:
Let X = 18°
Therefore, 5X = 90°
⇒ 2X + 3X = 90°
⇒ 2X = 90° – 3X
Taking the sin on both sides, we get
sin 2X = sin (90˚ – 3X)
sin 2X = cos 3X [Since, sin (90 - y) = cos y] -------------- (1)
We know that, according to the formula of multiple angles,
sin2x = 2 cos x sin x ------------ (2)
cos3x = 4cos3x – 3cosx -------------- (3)
Substituting (2) and (3) in (1) we get,
⇒ 2 sin X cos X = 4 cos3X – 3 cos X
⇒ 2 sin X cos X – 4 cos3X + 3 cos X = 0
⇒ cos X (2 sin X – 4 cos2X + 3) = 0
Divide both sides by cos X we get,
⇒ 2 sin X – 4 cos2X + 3 = 0
⇒ 2sinX – 4 (1 – sin2X) + 3 = 0 [Since, sin2y + cos2y = 1]
⇒ 4sin2X + 2sinX – 1 = 0
Let's use the quadratic formula to calculate the roots of the above quadratic equation
For a quadratic equation ax2 +bx +c = 0, the values of root x will be
x = [−b ± √(b2−4ac)] / 2a
In the equation, 4sin2X + 2sinX – 1 = 0 we have
a = 4, b = 2, c = -1
Therefore,
sin X = [−2 ± √22 −4(4)(−1)] / 2(4)
⇒ sinX = [−2 ± √4+16] / 8
⇒ sinX = [−2 ± 2√5] / 8
⇒ sinX = (−1 ± √5) / 4
We will consider sinX = (−1 + √5) /4 because sin 18° lies in the first quadrant and it is always grater than 0.
Therefore,
sin 18° = sin X = −1 + √5/4
Now, cos 36° = cos (2×18°)
⇒ cos 36° = 1 – 2 sin218° [ Since, cos 2x = 1 - 2 sin2x]
⇒ cos 36° = 1 – 2[(−1 + √5) / 4]2
⇒ cos 36° = [16 − 2 (5+1−2√5)] /16
⇒ cos 36° = (4 + 4√5) /16
⇒ cos 36° = (√5+1) /4
Thus, cos 36 = (√5+1) /4
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