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Course: Digital SAT Math > Unit 13

Lesson 3: Right triangle trigonometry: advanced
Test prep
Digital SAT Math
Advanced: Geometry and trigonometry
Right triangle trigonometry: advanced
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Right triangle trigonometry | Lesson

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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

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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:
A right triangle has leg lengths of a and b and hypotenuse length of c.
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!
In right triangle ABC, A is the right angle. The length of leg AB is 4, and the length of leg AC is 6.
What is the length of BC― in the figure above?
In right triangle DEF, F is the right angle. The length of leg DF is 24, and the length of hypotenuse DE is 25.
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:
A right triangle has leg lengths of 3 and 4 and hypotenuse length of 5.
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
Triangle ACD has right angle D, base AD, and height CD. B is a point on AC, E is a point on AD, and line segment BE divides the triangle ACD into a smaller triangle on the left and a quadrilateral on the right. AB has length 5, and AE has length 4.
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

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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.
Right triangle ABC has hypotenuse AB, longer leg BC, and shorter leg AC. The measure of angle A is theta, and angle C is the right angle.
For right triangle ABC with angle θ shown above:
sin⁡θ=opposite leghypotenuse=BCABcos⁡θ=adjacent leghypotenuse=ACABtan⁡θ=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
Right triangles ABC and DEF are shown, For right triangle ABC, angle A is the right angle, AB=4, AC=3, and BC=5. Right triangle DEF is larger than right triangle ABC, and angle D is the right angle. DE is the longer leg, and DF is the shorter leg.
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 right triangle ABC, A is the right angle, AB is the shorter leg, AC is the longer leg, and BC is the hypotenuse. AB=8.
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:
A right triangle has acute angles measuring 30 degrees and 60 degrees. The shorter leg of the triangle is opposite of the 30-degree angle and has length x. The longer leg of the triangle is opposite of the 60-degree angle and has length x times the square root of 3. The hypotenuse of the triangle has length 2x.
A right triangle has two acute angles each measuring 45 degrees. Both legs of the triangle have length s, and the hypotenuse of the triangle has length s times the square root of 2.
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 has right angle B, longer leg AB, shorter leg BC, and hypotenuse AC. The length of BC is 5.
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

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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 90∘−θ to represent the measure of angle B.
Right triangle ABC has hypotenuse AB, longer leg BC, and shorter leg AC. The measure of angle A is theta, the measure of angle B is 90 minus theta, and angle C is the right angle.
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
Right triangle ACE has right angle A, base AE, and height AC. B is a point on AC, D is a point on CE, and line segment BD is parallel to AE. The length of base AE is 12, the length of line segment BD is 8, and the length of line segment BC is 6.
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
Two right triangles, ABC and DEF, are shown. For triangle ABC, A is the right angle, AB=7, BC=25, and AC=24. For triangle DEF, D is the right angle, DE is the shorter leg, EF is the hypotenuse, and DF is the longer leg. Triangle DEF is smaller than triangle ABC.
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
Right triangle ABC has base AC and right angle B. Point D is on AC, and BD is the height of triangle ABC. The length of AB is 15, and the length of AD is 9.
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
Quadrilateral ABCD has parallel horizontal sides AD and BC, with BC shorter than AD. Side AB is vertical and perpendicular to both sides AD and BC.
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 cos⁡x∘=513. What is the the value of sin⁡(90∘−x∘) ?
  • 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

sin⁡θ=opposite leghypotenusecos⁡θ=adjacent leghypotenusetan⁡θ=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
sin⁡θ=cos⁡(90∘−θ)

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