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.

## SAT (Fall 2023)

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

Lesson 6: Additional Topics in Math: lessons by skill

# 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:
1. Find the sine, cosine, and tangent of similar triangles
2. 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

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 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: 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: 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 $\text{SOHCAHTOA}$:
• 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?

• should we learn all the values for sin, cos, and tan functions? • 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).  