Finding Missing Sides with Trigonometry

One of the most practical skills in trigonometry is using an angle and one known side to find an unknown side of a right triangle. This is the core technique carpenters use for rafter calculations, electricians use for conduit runs, and surveyors use for distance measurements.

The process is always the same: identify the sides, choose the right ratio, set up the equation, and solve.

The Three-Step Method

Step 1 — Label the sides. Relative to the known angle, identify which sides are the opposite (O), adjacent (A), and hypotenuse (H).

Step 2 — Choose the ratio. Based on which side you know and which side you need, pick sine, cosine, or tangent:

Step 3 — Solve. Plug in the values, then use algebra to isolate the unknown.

Finding the Opposite Side (Given Angle and Hypotenuse)

When you know the hypotenuse and an angle, use sine to find the opposite side.

Example 1: A cable is 40 feet long and makes a 35-degree angle with the ground. How high above the ground does the cable reach?

The cable is the hypotenuse ($H = 40$). The height is opposite the 35-degree angle. We need $O$.

$\sin(35°) = \frac{O}{40}$

$O = 40 \times \sin(35°) \approx 40 \times 0.5736 \approx 22.94 \text{ ft}$

Answer: The cable reaches approximately 22.94 feet above the ground.

Finding the Adjacent Side (Given Angle and Opposite)

When you know the opposite side and an angle, use tangent to find the adjacent side.

Example 2: A tree casts a shadow. From the tip of the shadow, the angle up to the top of the tree is 50 degrees. The tree is 30 feet tall. How long is the shadow?

The tree height (30 ft) is opposite the 50-degree angle. The shadow length is adjacent. We need $A$.

$\tan(50°) = \frac{O}{A} = \frac{30}{A}$

$A = \frac{30}{\tan(50°)} \approx \frac{30}{1.1918} \approx 25.17 \text{ ft}$

Answer: The shadow is approximately 25.17 feet long.

Finding the Hypotenuse (Given Angle and One Side)

Example 3: Find the hypotenuse given an angle and the adjacent side.

A ramp makes a 12-degree angle with the ground. The horizontal run is 20 feet. How long is the ramp surface?

The horizontal run (20 ft) is adjacent to the 12-degree angle. The ramp surface is the hypotenuse. We need $H$.

$\cos(12°) = \frac{A}{H} = \frac{20}{H}$

$H = \frac{20}{\cos(12°)} \approx \frac{20}{0.9781} \approx 20.45 \text{ ft}$

Answer: The ramp surface is approximately 20.45 feet long.

Solving Tips

When the unknown is in the numerator of the ratio, multiply:

$\sin\theta = \frac{O}{H} \implies O = H \times \sin\theta$

When the unknown is in the denominator, divide:

$\sin\theta = \frac{O}{H} \implies H = \frac{O}{\sin\theta}$

Here is the complete set of rearranged formulas:

Real-World Application: Carpentry — Rafter Length Calculation

A carpenter is framing a roof. The roof pitch angle is 25 degrees, and the horizontal run from the outer wall to the ridge beam is 14 feet. The carpenter needs to know the rafter length (the actual board that runs along the slope from wall to ridge).

The horizontal run (14 ft) is adjacent to the 25-degree pitch angle. The rafter length is the hypotenuse.

Step 1 — Choose the ratio. We know Adjacent and need Hypotenuse. Use cosine.

$\cos(25°) = \frac{14}{H}$

Step 2 — Solve for $H$:

$H = \frac{14}{\cos(25°)} \approx \frac{14}{0.9063} \approx 15.45 \text{ ft}$

Step 3 — Find the rise (for ordering materials). The rise is the opposite side:

$O = 14 \times \tan(25°) \approx 14 \times 0.4663 \approx 6.53 \text{ ft}$

Answer: The rafter needs to be at least 15.45 feet long, and the roof will rise 6.53 feet from the wall plate to the ridge. The carpenter would round up to 16 feet for the lumber order and add extra for the overhang.

Common Mistakes

1. Using the wrong ratio. If you need the relationship between opposite and hypotenuse, use sine — not cosine or tangent.
2. Multiplying when you should divide. When the unknown side is in the denominator of the ratio, you need to divide, not multiply.
3. Forgetting to check with Pythagorean theorem. After finding all three sides, verify with $a^2 + b^2 = c^2$.
4. Rounding too early. Keep at least 4 decimal places in intermediate steps and round only in the final answer.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• To find a missing side, label the sides (O, A, H), choose the ratio that connects your known and unknown sides, and solve
• When the unknown is in the numerator, multiply: $O = H \sin\theta$
• When the unknown is in the denominator, divide: $H = O / \sin\theta$
• Always verify your answer using the Pythagorean theorem when possible
• This technique is used daily in carpentry (rafter lengths), electrical work (conduit runs), and surveying (indirect measurement)

Return to Trigonometry for more topics in this section.