If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Digital SAT Math>Unit 13

Lesson 3: Right triangle trigonometry: advanced

# 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

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:
${c}^{2}={a}^{2}+{b}^{2}$

#### 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 $\stackrel{―}{BC}$ in the figure above?
What is the length of $\stackrel{―}{EF}$ 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 ${5}^{2}={3}^{2}+{4}^{2}$. $3$-$4$-$5$ 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 $7$-$24$-$25$.

### Try it!

try: use pythagorean triples and similarity to find side lengths
In the figure above, $\stackrel{―}{BE}$ is parallel to $\stackrel{―}{CD}$.
What is the length of $\stackrel{―}{BE}$ ?
If $CD=6$, what is the length of $\stackrel{―}{AC}$ ?

## What are the trigonometric ratios?

### Triangle similarity & trigonometric ratios

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 $\theta$ and side lengths in a right triangle.
For right triangle $ABC$ with angle $\theta$ shown above:
$\begin{array}{rl}\mathrm{sin}\theta & =\frac{\text{opposite leg}}{\text{hypotenuse}}=\frac{BC}{AB}\\ \\ \mathrm{cos}\theta & =\frac{\text{adjacent leg}}{\text{hypotenuse}}=\frac{AC}{AB}\\ \\ \mathrm{tan}\theta & =\frac{\text{opposite leg}}{\text{adjacent leg}}=\frac{BC}{AC}\end{array}$
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.

### Try it!

try: find the trigonometric ratios for two similar triangles
In the figure above, triangles $ABC$ and $DEF$ are similar.
What is $\mathrm{cos}\left(C\right)$ ? Enter your answer as a fraction.
Which angle in triangle $DEF$ has the same measure as angle $C$ in triangle $ABC$ ?
What is $\mathrm{tan}\left(F\right)$ ? Enter your answer as a fraction.

## 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, $\mathrm{tan}\left(C\right)=\frac{4}{7}$. What is the length of $\stackrel{―}{AC}$ ?

#### 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 ${30}^{\circ }$-${60}^{\circ }$-${90}^{\circ }$ triangle and the length of the shorter leg is $3$, we know that the length of the hypotenuse is $2\left(3\right)=6$ 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.

### Try it!

try: recognize trigonometric ratios and special right triangles
Right triangle $ABC$ is shown in the figure above. The value of $\mathrm{sin}\left(A\right)$ is $\frac{1}{2}$.
What is the length of $\stackrel{―}{AC}$ ?
What is the measure of angle $C$ ?
${}^{\circ }$

## How are the sine and cosine of complementary angles related?

### Sine & cosine of complementary angles

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 $\theta$ to represent the measure of angle $A$, we can use ${90}^{\circ }-\theta$ to represent the measure of angle $B$.
We can show that $\mathrm{sin}\left(A\right)=\mathrm{cos}\left(B\right)$. The hypotenuse, $\stackrel{―}{AB}$, is the same for both angles. However, $\stackrel{―}{BD}$ is opposite to angle $A$ but adjacent to angle $B$.
$\begin{array}{rl}\mathrm{sin}\left(A\right)& =\frac{\text{opposite}}{\text{hypotenuse}}\\ \\ & =\frac{BC}{AB}\\ \\ \mathrm{cos}\left(B\right)& =\frac{\text{adjacent}}{\text{hypotenuse}}\\ \\ & =\frac{BC}{AB}\end{array}$

### 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.

Practice: find segment length
In the figure above, $\stackrel{―}{BD}$ is parallel to $\stackrel{―}{AE}$. What is the length of $\stackrel{―}{DE}$ ?

Practice: identify equivalent side length ratios
In the figure above, triangle $ABC$ is similar to triangle $DEF$. What is the value of $\mathrm{sin}\left(F\right)$ ?

Practice: find side length
In the figure above, $\mathrm{sin}\left(C\right)=\frac{3}{5}$. What is the length of $\stackrel{―}{BC}$ ?

Practice: find angle measure
In quadrilateral $ABCD$ above, $\stackrel{―}{AD}$ is parallel to $\stackrel{―}{BC}$ and $CD=\sqrt{2}AB$. What is the measure of angle $C$ ?

Practice: use the relationship between the sine and cosine of complementary angles
In a right triangle, one angle measures ${x}^{\circ }$, where $\mathrm{cos}{x}^{\circ }=\frac{5}{13}$. What is the the value of $\mathrm{sin}\left({90}^{\circ }-{x}^{\circ }\right)$ ?

## Things to remember

$\begin{array}{rl}\mathrm{sin}\theta & =\frac{\text{opposite leg}}{\text{hypotenuse}}\\ \\ \mathrm{cos}\theta & =\frac{\text{adjacent leg}}{\text{hypotenuse}}\\ \\ \mathrm{tan}\theta & =\frac{\text{opposite leg}}{\text{adjacent leg}}\end{array}$
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
$\mathrm{sin}\theta =\mathrm{cos}\left({90}^{\circ }-\theta \right)$

## Want to join the conversation?

• Lord have mercy...
• Christ have mercy and give me 1600
• hello good luck everyone
• wish you the same!
• there was mistakenly typed BD instead of BC.
• Hi! are sec/cosec/cot included on the digital SAT?
• Hey, no they aren't included. Only sin, cos, and tan functions are included.
• going to cry because the ending made no sense T^T everyone wish me luck sob sob