Trigonometric Ratios

Trigonometry connects the angles of a triangle to the lengths of its sides. In a right triangle, the three basic trigonometric ratios—sine, cosine, and tangent—relate an acute angle to the ratio of two specific side lengths.

Before writing the ratios, we need to label the three sides of a right triangle relative to a specific acute angle, let's call it $\theta$:

• Hypotenuse: The longest side, always opposite the $90^\circ$ right angle.
• Opposite: The side directly across from the angle $\theta$.
• Adjacent: The side next to the angle $\theta$ that is not the hypotenuse.

The Three Basic Ratios (SOH CAH TOA)

A standard memory trick for these fundamental ratios is SOH CAH TOA:

• SOH: $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• CAH: $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• TOA: $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

These ratios allow you to find unknown side lengths or acute angle measures, but they apply only to the acute angles inside a right triangle.

Example 1: Finding Ratios from Side Lengths

Problem: In a right triangle with a hypotenuse of $13$ and one leg of $5$, find $\sin \theta$, $\cos \theta$, and $\tan \theta$ for the angle opposite the leg of length $5$.

Solution: First, we need to find the missing side (the adjacent leg) using the Pythagorean theorem ($a^2 + b^2 = c^2$): $5^2 + b^2 = 13^2$ $25 + b^2 = 169 \implies b^2 = 144 \implies b = 12$

Now, identify the sides relative to angle $\theta$:

• Opposite = $5$
• Adjacent = $12$
• Hypotenuse = $13$

Using SOH CAH TOA, we can write the ratios:

• $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{5}{13}$
• $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{12}{13}$
• $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{5}{12}$

Example 2: Finding Ratios from Another Ratio

Problem: If $\sin A = \frac{3}{5}$, find $\cos A$ and $\tan A$.

Solution: Since $\sin A = \frac{\text{Opposite}}{\text{Hypotenuse}}$, we can sketch a right triangle where the side opposite to angle $A$ is $3$ and the hypotenuse is $5$.

Find the adjacent side using the Pythagorean theorem: $a^2 + 3^2 = 5^2$ $a^2 + 9 = 25 \implies a^2 = 16 \implies a = 4$

So, the adjacent side is $4$. Now, write the remaining ratios using the side lengths $3$, $4$, and $5$:

• $\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5}$
• $\tan A = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{3}{4}$