Finding Missing Sides: Step-by-Step Guide
In a right triangle, if you know one acute angle and one side, you can find the other sides using SOH-CAH-TOA. Sine, cosine, and tangent are ratios (fractions) of the sides. Pick the one that connects the side you know with the side you want, then solve for the unknown.
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Label the sides. Using the angle you know, label the three sides: opposite (across from the angle), adjacent (next to the angle), and hypotenuse (across from the right angle).\[O,\,A,\,H\]
- Pick the right ratio. SOH: sin = opposite / hypotenuse. CAH: cos = adjacent / hypotenuse. TOA: tan = opposite / adjacent. Choose the one that has the side you know and the side you want.\[\sin\theta = \frac{O}{H},\;\cos\theta = \frac{A}{H},\;\tan\theta = \frac{O}{A}\]
- Write the equation and solve. Put the angle and the known side in, then solve for the unknown side.\[x = H \sin\theta\;\text{or}\; x = \frac{O}{\sin\theta}\]
- Use a calculator and round. Make sure your calculator is in degree mode. Round at the end.\[x \approx \text{value}\]
- Check your answer. Use the Pythagorean theorem or another ratio to see if the sides fit.\[a^2 + b^2 = c^2\]
Worked example
In a right triangle, the hypotenuse is 10 and one acute angle is 30 degrees. Find the side opposite the 30-degree angle.
Step 1: The hypotenuse is 10 and the angle is 30 degrees. We want the side opposite the angle. Call it x.
Step 2: We know the hypotenuse and want the opposite side, so we use sine (SOH).
Step 3: sin(30 degrees) = x / 10, so x = 10 x sin(30 degrees).
Step 4: sin(30 degrees) = 1/2, so x = 10 x (1/2) = 5.
Step 5: Check: the adjacent side is 10 x cos(30 degrees) = 5 x the square root of 3, which is about 8.66. Then 5^2 + (5 x the square root of 3)^2 = 25 + 75 = 100 = 10^2. It works.
The opposite side is 5.
Tips and common mistakes
- Always begin by labeling the sides from the point of view of the angle you know.
- If the unknown side is on top of the fraction, multiply. If it is on the bottom, divide.
- Make sure your calculator is in degree mode, not radian mode.
- After you find two sides, use the Pythagorean theorem as a check.
- 30, 45, and 60 degrees have exact values you can memorize, like sin(30 degrees) = 1/2.
- Do not round in the middle of a problem. Round only at the end.
- If you know two angles and one side, you can also use the Law of Sines.
- Draw the triangle and label every side and angle so you do not mix them up.