SAT Math 14: Right Triangles and Trigonometry

Why do trigonometric ratios depend only on an angle in a right triangle?

Right triangles with the same acute angle are similar, so corresponding side-length ratios are constant. Therefore, each ratio defines a function of the angle.

In a right triangle, how are sine, cosine, and tangent defined for an acute angle $A$?

$\sin A=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos A=\frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan A=\frac{\text{opposite}}{\text{adjacent}}$.

SOH-CAH-TOA

A mnemonic for right-triangle ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent.

How are tangent, sine, and cosine related?

$\tan A=\frac{\sin A}{\cos A}$, wherever $\cos A\ne 0$.

What is the hypotenuse of a right triangle?

It is the side opposite the $90^\circ$ angle and is the longest side of the triangle.

What are the exact values of $\sin\theta$ and $\cos\theta$ at $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, and $\frac{\pi}{2}$?

For these angles, $\sin\theta$ values are $0$, $\frac{1}{2}$, $\frac{\sqrt2}{2}$, $\frac{\sqrt3}{2}$, and $1$; $\cos\theta$ values are $1$, $\frac{\sqrt3}{2}$, $\frac{\sqrt2}{2}$, $\frac{1}{2}$, and $0$, respectively.

How are degrees and radians related?

$180^\circ=\pi$ radians, so $\theta_{\text{rad}}=\theta_{\text{deg}}\frac{\pi}{180}$ and $\theta_{\text{deg}}=\theta_{\text{rad}}\frac{180}{\pi}$.

How can trigonometric functions be used to determine an unknown side or angle in a right triangle?

Choose the ratio containing the known and unknown quantities, substitute the side lengths, and solve. If an angle is unknown, use the corresponding inverse function; for example, $A=\arctan\left(\frac{\text{opposite}}{\text{adjacent}}\right)$.

Inverse sine, or $\arcsin x$

$y=\arcsin x$ means $\sin y=x$. For real inputs, $-1\le x\le1$, and the principal output lies in $[-\frac{\pi}{2},\frac{\pi}{2}]$.

Inverse cosine, or $\arccos x$

$y=\arccos x$ means $\cos y=x$. For real inputs, $-1\le x\le1$, and the principal output lies in $[0,\pi]$.

Inverse tangent, or $\arctan x$

$y=\arctan x$ means $\tan y=x$. Any real number is allowed as input, and the principal output lies in $(-\frac{\pi}{2},\frac{\pi}{2})$.

What is the Pythagorean trigonometric identity?

$\sin^2\theta+\cos^2\theta=1$. In a right triangle, it is related to the Pythagorean theorem.

How can the other two Pythagorean trigonometric identities be obtained?

Dividing $\sin^2\theta+\cos^2\theta=1$ by $\cos^2\theta$ gives $\tan^2\theta+1=\sec^2\theta$; dividing by $\sin^2\theta$ gives $1+\cot^2\theta=\csc^2\theta$.

What are the side-length relationships in a $30$-$60$-$90$ triangle?

The side lengths are in the ratio $1:\sqrt{3}:2$. The side opposite $30^\circ$ is the shortest side, the side opposite $60^\circ$ is $\sqrt{3}$ times the shortest side, and the hypotenuse is twice the shortest side.

What are the side-length relationships in a $45$-$45$-$90$ triangle?

The two legs are congruent. If each leg has length $x$, the hypotenuse has length $x\sqrt{2}$, so the side ratio is $1:1:\sqrt{2}$.

How can a $30$-$60$-$90$ triangle be used to find exact trigonometric values?

Using side ratio $1:\sqrt{3}:2$, $\sin30^\circ=\frac{1}{2}$, $\cos30^\circ=\frac{\sqrt3}{2}$, $\tan30^\circ=\frac{\sqrt3}{3}$, and the corresponding values for $60^\circ$ are $\frac{\sqrt3}{2}$, $\frac{1}{2}$, and $\sqrt3$.

How can a $45$-$45$-$90$ triangle be used to find exact trigonometric values?

Because its legs are equal and its hypotenuse is $\sqrt2$ times a leg, $\sin45^\circ=\cos45^\circ=\frac{\sqrt2}{2}$ and $\tan45^\circ=1$.

What happens to sine, cosine, and tangent under a complementary-angle substitution?

For complementary angles, $\sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta$ and $\cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta$. Similarly, $\tan\left(\frac{\pi}{2}-\theta\right)=\cot\theta$ when both sides are defined.

How is trigonometry used in triangulation?

Triangulation uses measured angles and a known baseline to calculate unknown distances or positions through relationships among the sides and angles of triangles. It is used in surveying, navigation, astronomy, and positioning systems.