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SAT

Course: SAT>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 A, B, C with angle theta shown above:
\begin{aligned} \sin \theta &=\dfrac{\text{opposite leg}}{\text{hypotenuse}}=\dfrac{\maroonD{BC}}{\purpleD{AB}} \\\\ \cos \theta &=\dfrac{\text{adjacent leg}}{\text{hypotenuse}} =\dfrac{\tealE{AC}}{\purpleD{AB}}\\\\ \tan \theta &=\dfrac{\text{opposite leg}}{\text{adjacent leg}}=\dfrac{\maroonD{BC}}{\tealE{AC}} \end{aligned}
A common way to remember the trigonometric ratios is the mnemonic start text, S, O, H, C, A, H, T, O, A, end text:
• 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 A, B, C and D, E, F are similar.
What is cosine, left parenthesis, C, right parenthesis ? Enter your answer as a fraction.
Which angle in triangle D, E, F has the same measure as angle C in triangle A, B, C ?
What is tangent, left parenthesis, F, right parenthesis ? 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, degrees, minus, theta to represent the measure of angle B.
We can show that sine, left parenthesis, A, right parenthesis, equals, cosine, left parenthesis, B, right parenthesis. The hypotenuse, start color #7854ab, start overline, A, B, end overline, end color #7854ab, is the same for both angles. However, start color #ca337c, start overline, B, D, end overline, end color #ca337c is opposite to angle A but adjacent to angle B.
\begin{aligned} \sin(A) &=\dfrac{\text{opposite}}{\text{hypotenuse}} \\\\ &=\dfrac{\maroonD{BC}}{\purpleD{AB}} \\\\ \cos (B) &=\dfrac{\text{adjacent}}{\text{hypotenuse}} \\\\ &=\dfrac{\maroonD{BC}}{\purpleD{AB}} \end{aligned}

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 A, B, C is similar to triangle D, E, F. What is the value of sine, left parenthesis, F, right parenthesis ?

Practice: use the relationship between the sine and cosine of complementary angles
In a right triangle, one angle measures x, degrees, where cosine, x, degrees, equals, start fraction, 5, divided by, 13, end fraction. What is the the value of sine, left parenthesis, 90, degrees, minus, x, degrees, right parenthesis ?

Things to remember

\begin{aligned} \sin \theta &=\dfrac{\text{opposite leg}}{\text{hypotenuse}} \\\\ \cos \theta &=\dfrac{\text{adjacent leg}}{\text{hypotenuse}} \\\\ \tan \theta &=\dfrac{\text{opposite leg}}{\text{adjacent leg}} \end{aligned}
A common way to remember the trigonometric ratios is the mnemonic start text, S, O, H, C, A, H, T, O, A, end text:
• Sine is Opposite over Hypotenuse
• Cosine is Adjacent over Hypotenuse
• Tangent is Opposite over Adjacent
sine, theta, equals, cosine, left parenthesis, 90, degrees, minus, theta, right parenthesis

Want to join the conversation?

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