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### 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:

- Use the Pythagorean theorem and recognize Pythagorean triples
- Find the sine, cosine, and tangent of similar triangles
- Use trigonometric ratios to calculate side lengths
- Recognize special right triangles and use them to find side lengths and angle measures
- 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

### 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:

#### 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 $\stackrel{\u2015}{BC}$ in the figure above?

What is the length of $\stackrel{\u2015}{EF}$ in the figure above?

#### Recognizing Pythagorean triples

**Pythagorean triples**are integers

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!

## What are the trigonometric ratios?

### Triangle similarity & trigonometric ratios

### 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:

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

**S**ine is**O**pposite over**H**ypotenuse**C**osine is**A**djacent over**H**ypotenuse**T**angent is**O**pposite over**A**djacent

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!

## 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, $\mathrm{tan}(C)={\displaystyle \frac{4}{7}}$ . What is the length of $\stackrel{\u2015}{AC}$ ?

#### 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 $2(3)=6$ 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.

### Try it!

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

### Sine & cosine of complementary angles

### 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 $\mathrm{sin}(A)=\mathrm{cos}(B)$ . The hypotenuse, ${\stackrel{\u2015}{AB}}$ , is the same for both angles. However, ${\stackrel{\u2015}{BD}}$ is opposite to angle $A$ but adjacent to angle $B$ .

### Try it!

## Your turn!

## Things to remember

A common way to remember the trigonometric ratios is the mnemonic

**SOHCAHTOA**:**S**ine is**O**pposite over**H**ypotenuse**C**osine is**A**djacent over**H**ypotenuse**T**angent is**O**pposite over**A**djacent

## Want to join the conversation?

- Lord have mercy...(226 votes)
- Christ have mercy and give me 1600(67 votes)

- hello good luck everyone(49 votes)
- wish you the same!(22 votes)

- there was mistakenly typed BD instead of BC.(56 votes)
- Hi! are sec/cosec/cot included on the digital SAT?(17 votes)
- Hey, no they aren't included. Only sin, cos, and tan functions are included.(54 votes)

- going to cry because the ending made no sense T^T everyone wish me luck sob sob(37 votes)
- How to distinguish opposite leg and adjacent leg for once and forever? I've memorized the formula but still troubling with labeling one as opposite or adjacent(9 votes)
- basically the opposite is the one the angle isn't touching, and the adjacent literally means right next to. so opposite = not touching, and adjacent = touching. this tip really helped me and I hope it helps you too!(22 votes)

- i am not an american and i am an international test taker, i learnt these things in highschool but with a little bit different names! how can i learn all of these? is there any resources i can learn these from? or i should just learn them while im learning them?(9 votes)
- Do we have to memorize the two special triangles?!(7 votes)
- Crying as we speak...(6 votes)
- easy peasy lemon squeezy(5 votes)