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Right triangle trigonometry | Lesson

A guide to right triangle trigonometry on the digital SAT

What are right triangle trigonometry problems?

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!

How do I calculate side lengths using the Pythagorean theorem?

Intro to the Pythagorean theorem

Khan Academy video wrapper
Intro to the Pythagorean theoremSee video transcript

The Pythagorean theorem

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:
c2=a2+b2

Calculating missing side lengths in right triangles

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 BC in the figure above?
What is the length of EF in the figure above?

Recognizing Pythagorean triples

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 52=32+42. 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.

Try it!

try: use pythagorean triples and similarity to find side lengths
In the figure above, BE is parallel to CD.
What is the length of BE ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
If CD=6, what is the length of AC ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

What are the trigonometric ratios?

Triangle similarity & trigonometric ratios

Khan Academy video wrapper
Triangle similarity & the trigonometric ratiosSee video transcript

Sine, cosine, and tangent

For the SAT, we're expected to know the trigonometric ratios sine, cosine, and tangent. These ratios are based on the relationships between angle θ and side lengths in a right triangle.
For right triangle ABC with angle θ shown above:
sinθ=opposite leghypotenuse=BCABcosθ=adjacent leghypotenuse=ACABtanθ=opposite legadjacent leg=BCAC
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
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.

Try it!

try: find the trigonometric ratios for two similar triangles
In the figure above, triangles ABC and DEF are similar.
What is cos(C) ? Enter your answer as a fraction.
  • Your answer should be
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
Which angle in triangle DEF has the same measure as angle C in triangle ABC ?
Choose 1 answer:
What is tan(F) ? Enter your answer as a fraction.
  • Your answer should be
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4

How do I use trigonometric ratios and the properties of special right triangles to solve for unknown values?

Recognizing side length ratios

Using trigonometric ratios to find side lengths

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)=47. What is the length of AC ?

Using special right triangles to determine side lengths and angle measures

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 30-60-90 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 33.
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 45-45-90 special right triangle.

Try it!

try: recognize trigonometric ratios and special right triangles
Right triangle ABC is shown in the figure above. The value of sin(A) is 12.
What is the length of AC ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
What is the measure of angle C ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

How are the sine and cosine of complementary angles related?

Sine & cosine of complementary angles

Khan Academy video wrapper
Sine & cosine of complementary anglesSee video transcript

Relating the sine and cosine of complementary angles

In any right triangle, such as the one shown below, the two acute angles are
. If we use θ to represent the measure of angle A, we can use 90θ to represent the measure of angle B.
We can show that sin(A)=cos(B). The hypotenuse, AB, is the same for both angles. However, BD is opposite to angle A but adjacent to angle B.
sin(A)=oppositehypotenuse=BCABcos(B)=adjacenthypotenuse=BCAB

Try it!

try: match trigonometric ratios with the same value
In the table below, match each cosine to a sine with the same value without using a calculator.
1

Your turn!

Practice: find segment length
In the figure above, BD is parallel to AE. What is the length of DE ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Practice: identify equivalent side length ratios
In the figure above, triangle ABC is similar to triangle DEF. What is the value of sin(F) ?
Choose 1 answer:

Practice: find side length
In the figure above, sin(C)=35. What is the length of BC ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Practice: find angle measure
In quadrilateral ABCD above, AD is parallel to BC and CD=2AB. What is the measure of angle C ?
Choose 1 answer:

Practice: use the relationship between the sine and cosine of complementary angles
In a right triangle, one angle measures x, where cosx=513. What is the the value of sin(90x) ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a proper fraction, like 1/2 or 6/10
  • a simplified improper fraction, like 7/4
  • an improper fraction, like 10/7 or 14/8
  • an exact decimal, like 0.75

Things to remember

sinθ=opposite leghypotenusecosθ=adjacent leghypotenusetanθ=opposite legadjacent leg
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
sinθ=cos(90θ)

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