If 1 + sin2 θ = 3 sin θ cos θ, then prove that tan θ = 1 or ½.

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Updated on: September 25th, 2023

If 1 + sin2 θ = 3 sin θ cos θ, it is proved that tan θ = 1 or ½.

Given: 1 + sin2 θ = 3 sin θ cos θ

We know that: sin2 θ + cos2 θ = 1

Substituting the above value we get:

[sin2 θ + cos2 θ] + sin2 θ = 3 sin θ cos θ [As sin2 θ + cos2 θ = 1]

In simplification we get the:

2sin2 θ + cos2 θ = 3 sin θ cos θ

Divide both sides by cos2 θ

2sin2 θ/cos2 θ + cos2 θ/cos2 θ = 3 sin θ cos θ/cos2 θ

2tan2 θ + 1 = 3 sin θ/cos θ [As sin θ/cos θ = tan θ]

2tan2 θ + 1 = 3 tan θ

2tan2 θ – 3 tan θ + 1 = 0

2tan2 θ – 2tan θ – tan θ + 1 = 0

2 tan θ (tan θ – 1) – 1 (tan θ – 1) = 0

In simplification we get the:

(2 tan θ – 1) (tan θ – 1) = 0

tan θ = 1 or ½

Trigonometric Ratio

Trigonometric ratios are based on the value of the ratio of sides of a right-angled triangle and contain the values of all trigonometric functions. The trigonometric ratios of a given acute angle are the ratios of the sides of a right-angled triangle with respect to that angle.

The right triangle’s three sides are as follows:

• Hypotenuse (the longest side) (the longest side)
• Perpendicular (opposite side to the angle) (opposite side to the angle)
• Base (Adjacent side to the angle) (Adjacent side to the angle)

Summary:

If 1 + sin2 θ = 3 sin θ cos θ, then prove that tan θ = 1 or ½.

If 1 + sin2 θ = 3 sin θ cos θ, then it is proved that tan θ = 1 or ½. In a right triangle, the tangent of an angle is the ratio of the length of the adjacent side to the length of the opposite side. It is written as tan θ = opposite/adjacent.

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