If \(\tan\theta = 5/12\) and \(\theta\) is in Quadrant III, what is the value of \(\sin\theta\)?
\(-5/13\)
We are told \(\tan\theta=\dfrac{5}{12}\). By definition tangent is the ratio of the side opposite the angle to the side adjacent to it, so we can read off opposite \(=5\) and adjacent \(=12\) for a reference right triangle.
The hypotenuse comes from the Pythagoras theorem: \(\text{hyp}=\sqrt{5^{2}+12^{2}}=\sqrt{25+144}=\sqrt{169}=13\). So we are dealing with the well-known 5-12-13 triple.
Sine is opposite over hypotenuse, hence the bare magnitude is \(|\sin\theta|=\dfrac{5}{13}\). This fixes the number; only the sign is still to be decided.
The sign is set by the quadrant. Using the ASTC (All-Silver-Tea-Cups) rule, in Quadrant I all ratios are positive, in Quadrant II only sine, in Quadrant III only tangent, and in Quadrant IV only cosine.
Here \(\theta\) lies in Quadrant III, where only tangent is positive; both sine and cosine are negative. That is consistent with the given positive value of \(\tan\theta\).
Attaching the negative sign to the magnitude gives \(\sin\theta=-\dfrac{5}{13}\).
Key concept: build the numerical ratio from a right triangle (or from \(1+\tan^{2}\theta=\sec^{2}\theta\)), then attach the correct sign using the quadrant through the ASTC rule.
Hence, \(\sin\theta=\) \(-\dfrac{5}{13}\).
Similar Questions
A missile adjusts its flight path using a triangle formed with height = 7 units and hypotenuse = 25 units. If θ is the angle opposite to height and θ lies in the 1st quadrant, find tan(θ).
If sin x = 5/13 and x lies in (0°, 90°), then find the value of:
\(\dfrac{1 + \cos x}{1 - \cos x}\)
If sin θ + cos θ = $\frac{\sqrt3}{2}$, and θ is an acute angle, find the value of sin² θ + cos² θ − 2sin θ cos θ.
If in a right-angled triangle, cos A = 5/13, find the value of tan A – sin A.
If tan θ + cot θ = 5, and θ is an acute angle, find the value of tan² θ + cot² θ.
If \(\tan\theta = \frac{12}{5}\), find \(\sin\theta\).
What is the degree measure of an angle of \(\frac{7\pi}{4}\) radians?
If \(\cos A + \sin A = \dfrac{5}{4}\), find \(\tan A\).
If \(\sec^2\theta = 4\), then find the value of \(\sin^2\theta + \csc(90 - \theta)\) where \(0^\circ \lt \theta \lt 90^\circ\).
If \(\tan\theta + \cot\theta = 2\) and θ is an acute angle, find the value of \(\tan^3\theta + \cot^3\theta\).
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:
