Solving Right Triangles on the SAT
Find Missing Sides and Angles
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Solving Right Triangles
Solving a right triangle means finding missing sides or angles using trigonometric ratios. The SAT often hides these problems inside word problems about ladders, shadows, or ramps.
This lesson shows how to identify the triangle, choose the correct ratio, and solve for the missing value. You will also see when to use inverse trig and how to handle degree mode in Desmos.
A Simple Definition Unlocks Solving Right Triangles
Start by labeling the triangle. If you know an angle and one side, you can set up a sine, cosine, or tangent equation that connects the known side to the unknown side.
If you need the angle and you know two sides, use an inverse trig function. The SAT expects answers in degrees, so if you use Desmos, convert degrees to radians when typing or use the degree symbol in the input.
Work Through Solving Right Triangles Step by Step
In a right triangle with hypotenuse $10$ and angle $30^\circ$, what is the opposite side?
Find the opposite side when $\theta = 30^\circ$ and hypotenuse is $10$.
Use the sine ratio based on the given angle.
Use the exact value $\sin(30^\circ) = \frac{1}{2}$.
Solve for the opposite side
Use Desmos to Check Solving Right Triangles
Evaluate $\sin(42^\circ)$ as a decimal.
Desmos can compute trig values quickly. Use it when the angle is not a special angle and you need a decimal approximation.
sin(42 * pi/180)
Exact ratios are usually preferred on the SAT when special angles are involved.
Desmos is faster for messy decimals or when you need an inverse trig value quickly. Algebra is faster when the problem expects exact radicals or when the ratios are simple.
Expert move: Use Desmos to evaluate trig ratios or inverse trig when the angle is not special; keep exact values when possible and check the quadrant/sign.
- Desmos features used: trig evaluation.
- Common mistake: using the wrong ratio for the labeled sides.
Practice Solving Right Triangles with SAT-Style Questions
Solve each right triangle situation.
A right triangle has hypotenuse and adjacent side to angle . What is ?
If and the adjacent side is , what is the opposite side?
A right triangle has angle and hypotenuse . What is a leg?
If and hypotenuse is , what is the opposite side?
Key Takeaways to Remember for Solving Right Triangles
- Label sides relative to the angle, then choose the correct ratio.
- Use Desmos for decimals when the angle is not special.
- Check whether the question wants an exact or approximate value.
