Right triangle trigonometry | Lesson (article) | Khan Academy (2024)

Right triangle trigonometry problems are all about understanding the relationship between side lengths, angle measures, and trigonometric ratios in right triangles.

In this lesson, we'll learn to:

1. Use the Pythagorean theorem and recognize Pythagorean triples
2. Find the sine, cosine, and tangent of similar triangles
3. Use trigonometric ratios to calculate side lengths
4. Recognize special right triangles and use them to find side lengths and angle measures
5. Compare the sine and cosine of complementary angles

You can learn anything. Let's do this!

In a right triangle, the square of the hypotenuse length is equal to the sum of the squares of the leg lengths.

At the beginning of each SAT math section, you'll find this diagram provided as reference:

With the Pythagorean theorem, we can calculate any side length in a right triangle when given the other two.

Let's look at some examples!

What is the length of $\overline{BC}$‍ in the figure above?

What is the length of $\overline{EF}$‍ in the figure above?

Pythagorean triples are integers $a$‍, $b$‍, and $c$‍ that satisfy the Pythagorean theorem. For example, the side lengths of the right triangle shown below form a Pythagorean triple:

Each side of the triangle has an integer length, and $5^2=3^2+4^2$‍. $3$‍-$4$‍-$5$‍ is the most commonly used Pythagorean triple on the SAT. All triangles similar to it also have side lengths that are multiples of the $3$‍-$4$‍-$5$‍ Pythagorean triple, like $6$‍-$8$‍-$10$‍, $9$‍-$12$‍-$15$‍ or $30$‍-$40$‍-$50$‍.

Being able to recognize Pythagorean triples can save you valuable time on test day. For example, if you see a right triangle with a hypotenuse length of $15$‍ and a leg length of $12$‍, recognizing it's a $9$‍-$12$‍-$15$‍ triangle will give you the missing side length, $9$‍, without having to calculate it using the Pythagorean theorem.

Less frequently used Pythagorean triples include $5$‍-$12$‍-$13$‍ and $7$‍-$24$‍-$25$‍.

For the SAT, we're expected to know the trigonometric ratios sine, cosine, and tangent. These ratios are based on the relationships between angle $\theta$‍ and side lengths in a right triangle.

For right triangle $ABC$‍ with angle $\theta$‍ shown above:

A common way to remember the trigonometric ratios is the mnemonic $\text{SOHCAHTOA}$‍:

• Sine is Opposite over Hypotenuse
• Cosine is Adjacent over Hypotenuse
• Tangent is Opposite over Adjacent

Trigonometric ratios are constant for any given angle measure, which means corresponding angles in similar triangles have the same sine, cosine, and tangent. Therefore, if we can calculate the trigonometric ratios in one right triangle, we can also apply those ratios to similar triangles.

Sine, cosine, and tangent represent ratios of right triangle side lengths. This means if we have the value of the sine, cosine, or tangent of an angle and one side length, we can find the other side lengths.

Let's look at an example!

In the figure above, $\tan (C)={\displaystyle \frac{4}{7}}$‍. What is the length of $\overline{AC}$‍ ?

Special right triangles are right triangles with specific angle measure and side length relationships. At the beginning of each SAT math section, the following two special right triangles are provided as reference:

This means when we see a special right triangle with unknown side lengths, we know how the side lengths are related to each other. For example, if we have a ${30}^{\circ }$‍-${60}^{\circ }$‍-${90}^{\circ }$‍ triangle and the length of the shorter leg is $3$‍, we know that the length of the hypotenuse is $2(3)=6$‍ and the length of the longer leg is $3\sqrt{3}$‍.

We can also identify the angle measures of special right triangles when we spot specific side length relationships. For example, if we're given a right triangle with identical leg lengths, we know it's a ${45}^{\circ }$‍-${45}^{\circ }$‍-${90}^{\circ }$‍ special right triangle.

We can show that $\sin (A)=\cos (B)$‍. The hypotenuse, $\overline{AB}$‍, is the same for both angles. However, $\overline{BD}$‍ is opposite to angle $A$‍ but adjacent to angle $B$‍.

A common way to remember the trigonometric ratios is the mnemonic SOHCAHTOA:

• Sine is Opposite over Hypotenuse
• Cosine is Adjacent over Hypotenuse
• Tangent is Opposite over Adjacent