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Sine, Cosine, Tangent

Trigonometric ratios based on sides of right triangles in relation to an angle.

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Sine and Cosine Ratios

SOH-CAH-TOA is a mnemonic that many people use to remember the difference between sine, cosine, and tangent. How can remembering SOH-CAH-TOA help you?

Sine and Cosine Ratios

Two right triangles with one pair of non-right congruent angles are similar by \begin{align*}AA \sim\end{align*}. This means the ratio between the side lengths of the first triangle must be congruent to the ratio between the corresponding side lengths of the second triangle. 

For example, in the picture above, \begin{align*}\frac{a}{c}=\frac{d}{f}\end{align*}. Because there are three pairs of sides for any triangle, there are three relevant ratios for a given angle.

  1. The tangent of an angle gives the ratio \begin{align*}\frac{\text{opposite leg}}{\text{adjacent leg}}\end{align*}. The abbreviation for tangent is tan.
  2. The sine of an angle gives the ratio \begin{align*}\frac{\text{opposite leg}}{\text{hypotenuse}}\end{align*}. The abbreviation for sine is sin.
  3. The cosine of an angle gives the ratio \begin{align*}\frac{\text{adjacent leg}}{\text{hypotenuse}}\end{align*}. The abbreviation for cosine is cos.

These are the basic trigonometric ratios. Trigonometry is the study of triangles. These ratios are called trigonometric ratios because they apply to triangles. Just as your scientific or graphing calculator has tangent programmed into it, it also has sine and cosine programmed into it. This means that you can use your calculator to determine the ratio between the lengths of any pair of sides for any angle within a right triangle.

Finding Sine and Cosine Ratios 

Use your calculator to find the sine ratio and cosine ratio for a \begin{align*}27^\circ\end{align*} angle.

\begin{align*}\sin (27^\circ) \approx 0.454\end{align*} and \begin{align*}\cos (27^\circ) \approx 0.891\end{align*}.

Solving for Unknown Values 

Solve for \begin{align*}x\end{align*}.

Look to see how the sides that are marked are related to the \begin{align*}27^\circ\end{align*} angle. The side of length 11 is adjacent to the angle. The side of length \begin{align*}x\end{align*} is the hypotenuse of the triangle. When working with the adjacent side and the hypotenuse, you should use the cosine ratio.

\begin{align*}\cos (27^\circ) &= \frac{\text{adjacent leg}}{\text{hypothenuse}}\\ \cos (27^\circ) &= \frac{11}{x}\\ x &= \frac{11}{\cos (27^\circ)}\\ x & \approx \frac{11}{0.891}\\ x & \approx 12.346\end{align*}

Note that \begin{align*}\frac{11}{\cos (27^\circ)}\end{align*} is the exact answer. 12.346 is an approximate answer because you rounded the value of \begin{align*}\cos (27^\circ)\end{align*}.

Calculating Sine and Cosine Functions 

Find \begin{align*}\sin \theta\end{align*} and \begin{align*}\cos \theta\end{align*}.

Relative to angle \begin{align*}\theta\end{align*}, 3 is the opposite leg and 4 is the adjacent leg. In order to find the sine and cosine ratios you also need to know the hypotenuse of the triangle. Use the Pythagorean Theorem to find the hypotenuse:

\begin{align*}3^2+4^2 &= h^2\\ 25 &= h^2\\ h &= 5\end{align*}

Now, write the sine and cosine ratios:

\begin{align*}\sin \theta &= \frac{\text{opposite leg}}{\text{hypotenuse}}=\frac{3}{5}\\ \cos \theta &= \frac{\text{adjacent leg}}{\text{hypotenuse}}=\frac{4}{5}\end{align*}

Examples

Example 1

Earlier, you were asked how can remembering SOH-CAH-TOA can help you. 

SOH-CAH-TOA is a mnemonic that many people use to remember the difference between sine, cosine, and tangent. How can remembering SOH-CAH-TOA help you?

SOH-CAH-TOA stands for Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, and Tangent equals Opposite over Adjacent. This mnemonic helps you remember which trigonometric ratio is which.

Example 2

Use your calculator to find the sine and cosine ratios for a \begin{align*}39^\circ\end{align*} angle.

 \begin{align*}\sin (39^\circ) \approx 0.629\end{align*} and \begin{align*}\cos (39^\circ) \approx 0.777\end{align*}.

Example 3

Solve for \begin{align*}x\end{align*}.

Look to see how the sides that are marked are related to the \begin{align*}39^\circ\end{align*} angle. The side of length 17 is opposite from the angle. The side of length \begin{align*}x\end{align*} is the hypotenuse of the triangle. When working with the opposite side and the hypotenuse, you should use the sine ratio.

\begin{align*}\sin (39^\circ) &= \frac{\text{opposite leg}}{\text{hypotenuse}}\\ \sin (39^\circ) &= \frac{17}{x}\\ x &= \frac{17}{\sin (39^\circ)}\\ x & \approx \frac{11}{0.629}\\ x & \approx 27.013\end{align*}

Example 4

Find \begin{align*}\sin \theta\end{align*} and \begin{align*}\cos \theta\end{align*}.

Relative to angle \begin{align*}\theta\end{align*}, 4 is the opposite leg and 2 is the adjacent leg. In order to find the sine and cosine ratios you also need to know the hypotenuse of the triangle. Use the Pythagorean Theorem to find the hypotenuse:

\begin{align*}4^2+2^2 &= h^2\\ 20 &= h^2\\ h &= 2 \sqrt{5} \end{align*}

Now, write the sine and cosine ratios:

\begin{align*}\sin \theta &= \frac{\text{opposite leg}}{\text{hypotenuse}}=\frac{4}{2 \sqrt{5}}=\frac{2}{\sqrt{5}}=\frac{2 \sqrt{5}}{5}\\ \cos \theta &= \frac{\text{adjacent leg}}{\text{hypotenuse}}=\frac{2}{2 \sqrt{5}}=\frac{1}{\sqrt{5}}=\frac{\sqrt{5}}{5}\end{align*}

Note that in the last step of each calculation the denominator was rationalized. You can choose to rationalize the denominator if you wish.

Review

For #1-#6, use the triangle below. Find each exact value.

1. \begin{align*}\sin E\end{align*}

2. \begin{align*}\cos E\end{align*}

3. \begin{align*}\tan E\end{align*}

4. \begin{align*}\sin F\end{align*}

5. \begin{align*}\cos F\end{align*}

6. \begin{align*}\tan F\end{align*}

Identify whether the sine, cosine, or tangent ratio is most useful for helping to solve the problem. Then, solve for \begin{align*}x\end{align*}.

7.

8.

9.

10.

11.

12.

Use the triangle below for #13-#15.

13. Find \begin{align*}m \angle C\end{align*}.

14. Draw an altitude from \begin{align*}\angle B\end{align*} to divide the triangle into two right triangles. Use trigonometry to find the lengths of the sides of each of these right triangles.

15. Find the perimeter of \begin{align*}\Delta ABC\end{align*}.

16. True or false: You need to know the length of at least one side of a triangle to find the lengths of the other sides.

17. What are the trigonometric ratios?

18. What do the trigonometric ratios have to do with similar triangles?

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.2. 

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Vocabulary

TermDefinition
Acute Angle An acute angle is an angle with a measure of less than 90 degrees.
Adjacent Angles Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'next-to'.
Hypotenuse The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.
Legs of a Right Triangle The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.
opposite The opposite of a number x is -x. A number and its opposite always sum to zero.
Pythagorean Theorem The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by a^2 + b^2 = c^2, where a and b are legs of the triangle and c is the hypotenuse of the triangle.
Radical The \sqrt{}, or square root, sign.
sine The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse.
Trigonometric Ratios Ratios that help us to understand the relationships between sides and angles of right triangles.

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