Geometry Right Triangle Trigonometry Lesson (Free PDF)

Geometry math lesson: Right Triangle Trigonometry – Maths PDF

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This right triangles lesson for Geometry covers right triangle trigonometry (Common Core G-SRT.C.6, G-SRT.C.7, G-SRT.C.8), with a free PDF to print.

In a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. These ratios depend only on the acute angle because right triangles with the same angle are similar. Sine of an angle equals cosine of its complement. Use the ratios and the Pythagorean theorem to solve right triangles and applied problems.

Geometry · Common Core State Standards: G-SRT.C.6, G-SRT.C.7, G-SRT.C.8

Key facts in this right triangles lesson

Key facts

sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent.
sin A = cos (90 degrees minus A) for the acute angles of a right triangle.
Use an inverse function (sin^-1, cos^-1, tan^-1) to find an angle.
Solving a right triangle means finding all sides and acute angles.
Angles of elevation and depression are measured from the horizontal.

What are the trigonometric ratios?

In a right triangle, the hypotenuse is the side opposite the right angle. For an acute angle \(A\), the opposite side is across from it and the adjacent side touches it. Triangles with the same acute angles are similar, so these ratios depend only on the angle.

Sides are named relative to angle A.

Ratio	Definition	Memory aid
\(\text{sin}\,A\)	\(\frac{\text{opposite}}{\text{hypotenuse}}\)	SOH
\(\text{cos}\,A\)	\(\frac{\text{adjacent}}{\text{hypotenuse}}\)	CAH
\(\text{tan}\,A\)	\(\frac{\text{opposite}}{\text{adjacent}}\)	TOA

A 5-12-13 triangle: sin A = 5/13, cos A = 12/13, tan A = 5/12.

How are sine and cosine of complementary angles related?

The opposite side of one acute angle is the adjacent side of the other. So \(\text{sin}\,A = \text{cos}\,(90^{\circ} – A)\). For example, \(\text{sin}\,35^{\circ} = \text{cos}\,55^{\circ}\).

The acute angles are complementary: sin 35° = cos 55°.

How do you solve for a side or an angle?

Draw the triangle and mark the known angle and sides.
Choose the ratio that connects a known side and the unknown side (SOH, CAH or TOA).
Write an equation and solve. To find an angle, use an inverse function such as \(\text{tan}^{-1}\left( \frac{\text{opp}}{\text{adj}} \right)\).
Check with the Pythagorean theorem or the angle sum of 90°.

Use a ratio that connects the known side and the unknown side.

To solve a right triangle, find all three sides and both acute angles. Use the Pythagorean theorem \(a^{2} + b^{2} = c^{2}\) and the angle sum \(A + B = 90^{\circ}\) together with trig ratios.

What are angles of elevation and depression?

The angle of elevation is measured upward from the horizontal. The angle of depression is measured downward from the horizontal. Both create right triangles with the ground.

Angle of elevation: looking up from the horizontal.

Example: From 50 feet away the angle of elevation to the top of a tree is \(32^{\circ}\). Then \(\text{tan}\,32^{\circ} = \frac{h}{50}\), so \(h = 50 \cdot 0.6249 = 31.2\) feet.

Right triangles lesson: questions and answers

What does SOH-CAH-TOA mean?

It is a memory aid: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

Why do trig ratios depend only on the angle?

Right triangles with the same acute angle are similar by AA, so their side ratios are equal no matter how large the triangle is.

How do you find an angle when you know two sides?

Compute the ratio and use the inverse function. If the opposite side is 7 and the adjacent side is 24, the angle is tan inverse of 7/24, about 16.3 degrees.

How are sine and cosine of complementary angles related?

They are equal: sin of 25 degrees equals cos of 65 degrees, because the opposite side of one angle is the adjacent side of the other.

What is the difference between angle of elevation and angle of depression?

Elevation is measured upward from a horizontal line, depression downward from a horizontal line. The two are congruent alternate interior angles for the same line of sight.

How to use this right triangles lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

Download the right triangles lesson as a PDF with the button below and keep it with your notes.

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