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## SAT (Fall 2023)

### Course: SAT (Fall 2023) > Unit 6

Lesson 6: Additional Topics in Math: lessons by skill- Volume word problems | Lesson
- Right triangle word problems | Lesson
- Congruence and similarity | Lesson
- Right triangle trigonometry | Lesson
- Angles, arc lengths, and trig functions | Lesson
- Circle theorems | Lesson
- Circle equations | Lesson
- Complex numbers | Lesson

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# Right triangle trigonometry | Lesson

## What are right triangle trigonometry problems, and how frequently do they appear on the test?

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:

- Find the sine, cosine, and tangent of similar triangles
- Compare the sine and cosine of complementary angles

On your official SAT, you'll likely see

**1 question**that tests your understanding of right triangle trigonometry.This lesson builds upon the Congruence and similarity skill.

**Note:**This lesson is focused on recognizing trigonometric relationships in right triangles. To learn more about calculating angle measures and side lengths using the Pythagorean theorem, trigonometry, and special right triangles, check out the Right triangle word problems lesson.

**You can learn anything. Let's do this!**

## 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 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 $\text{SOHCAHTOA}$ :

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

## Want to join the conversation?

- should we learn all the values for sin, cos, and tan functions?(2 votes)
- In my opinion, it's really not too useful for the SAT to know every value of the trig ratios throughout the unit circle. You can memorize the values for sine and tangent between 0 and 90, and then use those to derive the rest.

Cosine is called the complementary function of sine, or the cosine of an angle is equal to the sine of the complementary angle. So, cos(60) = sin(30), cos(45) = sin(45), and so on. For the other quadrants, you can draw a little picture, and using your knowledge that sine is the vertical distance from the point on the unit circle to the origin and cosine is the horizontal, you can fill in the other angle values. And finally, cosine is negative where x is negative, sine is negative where y is negative, and tangent is negative where the slope of line containing the point on the unit circle and the origin is negative (Quadrants II and IV).(7 votes)

- I don't understand the second try(1 vote)
- The sine of an angle is opposite / hypotenuse. The cosine of that same angle would be the adjacent / hypotenuse. What this question is getting at is that the adjacent side for one angle is the opposite side for the angle on the other end of the triangle. Since we have a right triangle, those two angles would be complementary, or add up to 90 degrees.

We can write this in equation form as:

sin(x) = opposite / hypotenuse = cos(90 - x)

For this reason, sine and cosine are sometimes called complementary functions, because if you take the sine of a complementary angle you get the cosine of the normal angle, and vice versa.(5 votes)

- I used to hate doing trig problems. Then I started calculus and I miss when trig was the hardest part of math.(1 vote)