Right Triangles and Trigonometry

This skill covers right triangles and the basics of trigonometry: the Pythagorean theorem, the trig ratios sine, cosine, and tangent, the special right triangles ($45$-$45$-$90$ and $30$-$60$-$90$), the link between sine and cosine of complementary angles, and angles measured in radians on the unit circle.

• In a right triangle, the hypotenuse is the longest side and sits opposite the right angle.
• Each trig ratio compares two sides relative to a chosen acute angle: opposite, adjacent, and hypotenuse change when you switch angles.
• The two acute angles of a right triangle are complementary (they add to $90^\circ$), so the side opposite one is adjacent to the other. That is why $\sin x^\circ = \cos(90 - x)^\circ$.
• If you know one trig ratio, sketch a triangle with those side lengths, find the third side with the Pythagorean theorem, and read off any other ratio.
• Radians measure angles by arc length on a circle of radius $1$: $\pi$ radians $= 180^\circ$.

$a^2 + b^2 = c^2 \qquad \sin\theta = \frac{\text{opp}}{\text{hyp}} \quad \cos\theta = \frac{\text{adj}}{\text{hyp}} \quad \tan\theta = \frac{\text{opp}}{\text{adj}}$

• Common Pythagorean triples: $3$-$4$-$5$, $5$-$12$-$13$, $8$-$15$-$17$, $7$-$24$-$25$, and their multiples.
• Degrees to radians: multiply by $\frac{\pi}{180}$. So $60^\circ = \frac{\pi}{3}$ and $300^\circ = \frac{5\pi}{3}$.
• Unit circle: a point at angle $\theta$ is $(\cos\theta, \sin\theta)$. Sine is negative below the $x$-axis; cosine is negative left of the $y$-axis. Example: $\frac{5\pi}{3}$ is $300^\circ$, with reference angle $60^\circ$ in the fourth quadrant, so $\sin\frac{5\pi}{3} = -\frac{\sqrt{3}}{2}$.

A right triangle has legs $7$ and $24$. Find the hypotenuse, and find $\sin\theta$, $\cos\theta$, and $\tan\theta$ for the angle $\theta$ opposite the side of length $7$.

1. $c^2 = 7^2 + 24^2 = 49 + 576 = 625$, so $c = 25$.
2. Opposite $\theta$ is $7$, adjacent is $24$: $\sin\theta = \frac{7}{25}$, $\cos\theta = \frac{24}{25}$, $\tan\theta = \frac{7}{24}$.

Check: $\left(\frac{7}{25}\right)^2 + \left(\frac{24}{25}\right)^2 = \frac{49 + 576}{625} = 1$. Correct.

In right triangle $JKL$, angle $K$ is the right angle and $\tan J = \frac{8}{15}$. What is $\cos L$?

1. Relative to $J$: opposite side $KL = 8$, adjacent side $JK = 15$.
2. Hypotenuse: $JL = \sqrt{8^2 + 15^2} = \sqrt{289} = 17$.
3. Relative to $L$, the adjacent leg is $KL = 8$. So $\cos L = \frac{8}{17}$.

Check: $J$ and $L$ are complementary, so $\cos L = \sin J = \frac{8}{17}$. Correct.

• Reading sides from the wrong angle. "Opposite" and "adjacent" depend on which angle you are using; relabel when the question switches angles.
• Adding the wrong squares. The hypotenuse goes alone on one side: a leg is $\sqrt{c^2 - a^2}$, not $\sqrt{c^2 + a^2}$.
• Mixing up the $30$-$60$-$90$ legs. The short leg is opposite $30^\circ$; the leg opposite $60^\circ$ is $\sqrt{3}$ times it.
• Complementary vs. supplementary. $\sin x^\circ = \cos y^\circ$ means $x + y = 90$, not $180$.
• Ignoring the sign on the unit circle. $\sin\frac{4\pi}{3}$ is negative because the angle points below the $x$-axis.