If \(\tan\theta = \frac{12}{5}\), find \(\sin\theta\).
\(\frac{12}{13}\)
The tangent of an angle is the ratio of the side opposite the angle to the side adjacent to it, so \(\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{12}{5}\).
We can therefore model \(\theta\) as an angle in a right triangle whose opposite side is 12 and adjacent side is 5.
The hypotenuse follows from Pythagoras: \(\sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13\).
This is the well-known 5-12-13 Pythagorean triple.
Sine is the ratio of the opposite side to the hypotenuse, so \(\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{12}{13}\).
The key idea is that one ratio (tangent) fixes the two legs, and Pythagoras supplies the hypotenuse so every other ratio can be read off.
Hence \(\sin\theta = \dfrac{12}{13}\), i.e. 12/13.
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Important Questions from Trigonometric Ratios and Identities
What is the ratio of the greatest to the smallest value of 2 – 2 sin x – sin 2x, 0 ≤ x ≤ (π/2)?
If sinθ = \(\frac{4}{5}\) , Find the value of sin3θ
If x, y are acute angles, where 0 < x + y < 90° and sin(3x - 40°) = cos (3y + 40°), then the value of tan (x + y) is equal to
What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?
If \(\sin \theta =\frac{3}{5}\) and \(\cos \theta =\frac{4}{5}\) , then the value of \(\frac{1+\tan \theta}{1-\cot \theta}\) is:
