Question

If sin 2theta=cos 3theta and theta is acute angle then find the value of theta

Answer 1

Answer:

theta

theta

theta

theta

theta

Answer 2

Answer:

The value of θ is 18°.

Step-by-step explanation:

The formula of sin 2θ is:

sin 2θ = 2 sin θ cos θ

The formula of cos 3θ is:

cos 3θ = 4 cos³θ - 3 cos θ

It is provided that:

sin 2θ = cos 3θ

That is:

2 sin θ cos θ = 4 cos³θ - 3 cos θ

2 sin θ cos θ = cos θ (4 cos²θ - 3)

2 sin θ = 4 cos²θ - 3

2 sin θ = 4 (1 - sin²θ) - 3

2 sin θ = 4 - 4 sin²θ - 3

4 sin²θ + 2 sin θ - 1 = 0

The last equation is a quadratic equation.

Let x = sin θ.

The equation now is:

4 x² + 2 x - 1 = 0

a = 4

b = 2

c = -1

Compute the value of x as follows:

$x = \frac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a }\\x = \frac{ -2 \pm \sqrt{2^2 - 4(4)(-1)}}{ 2(4) }\\x = \frac{ -2 \pm \sqrt{4 - -16}}{ 8 }\\x = \frac{ -2 \pm \sqrt{20}}{ 8 }\\x = \frac{ -2 }{ 8 } \pm \frac{2\sqrt{5}\, }{ 8 }\\x = 0.309017, x = -0.809017$

If sin θ = 0.309017, then the value of θ is 18° approximately.

If sin θ = -0.809017, then the value of θ is -54° approximately.

Measure of an angle cannot be negative.

Thus, the value of θ is 18°.

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