Solving Right Triangles

To "solve" a right triangle means to find the exact measures of all its three sides and three angles. As long as you know the $90^\circ$ angle and at least two other pieces of information (like one side and one acute angle, or two sides), you can figure out the rest.

The Tools You Need

To solve a right triangle, you will rely on four main mathematical tools:

1. Trigonometric Ratios (SOH CAH TOA):
   • $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
   • $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
   • $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
2. Inverse Trigonometric Functions: Use $\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$ to find a missing angle when you know two sides.
3. Pythagorean Theorem: $a^2 + b^2 = c^2$, useful when you know two sides and need the third.
4. Triangle Angle Sum: All angles in a triangle add up to $180^\circ$. In a right triangle, the two acute angles always add up to $90^\circ$.

Example 1: Given One Side and One Angle

Problem: In right $\triangle ABC$, $\angle C = 90^\circ$, side $a = 8$, and $\angle A = 35^\circ$. Solve the triangle.

Step 1: Find the missing angle ($\angle B$). Since the acute angles must add up to $90^\circ$: $\angle B = 90^\circ - 35^\circ = 55^\circ$

Step 2: Find side $b$. We know $\angle A = 35^\circ$, the opposite side $a = 8$, and we want the adjacent side $b$. We use tangent: $\tan(35^\circ) = \frac{8}{b}$ $b = \frac{8}{\tan(35^\circ)} \approx \frac{8}{0.7002} \approx 11.43$

Step 3: Find the hypotenuse $c$. We can use sine (Opposite / Hypotenuse): $\sin(35^\circ) = \frac{8}{c}$ $c = \frac{8}{\sin(35^\circ)} \approx \frac{8}{0.5736} \approx 13.95$

The triangle is now solved: $\angle A = 35^\circ$, $\angle B = 55^\circ$, $\angle C = 90^\circ$, $a = 8$, $b \approx 11.43$, $c \approx 13.95$.

Example 2: Real-World Application

Trigonometry is incredibly useful for solving word problems involving heights and distances, often utilizing an angle of elevation (looking up) or an angle of depression (looking down).

Problem: A ladder $15$ ft long leans against a wall making a $70^\circ$ angle with the ground. How high up the wall does it reach?

Solution: Imagine the right triangle formed by the wall, the ground, and the ladder.

• The ladder is the hypotenuse: $c = 15$.
• The angle with the ground is $70^\circ$.
• The height on the wall is the side opposite the $70^\circ$ angle. Let's call it $h$.

Since we want the Opposite side and have the Hypotenuse, we use sine: $\sin(70^\circ) = \frac{h}{15}$ $h = 15 \cdot \sin(70^\circ)$ $h \approx 15 \cdot 0.9397 \approx 14.1 \text{ ft}$

The ladder reaches approximately $14.1$ feet up the wall.