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Sine, Cosine and Tangent and Their Inverse Functions

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Note! This entry is only valid for right triangles/right-angled triangles.

The sine, cosine, and tangent of an angle are more than just buttons on your calculator! These three functions describe the relationships between two sides in a right triangle.

The three different functions—sine, cosine, and tangent—exist because there are three possible relationships. When you study their formulas, it is useful to compare them to the figure.

Legs and hypothenuse marked on a right triangle

The relationship between the sides of a right triangle and the angles in the triangle is as follows:

Formula

Sine, Cosine, Tangent and Their Inverse Functions

\begin{array}{l} \sin B & = \frac{b}{c} B = \sin^{- 1} (\frac{b}{c}) \\ \cos B & = \frac{a}{c} B = \cos^{- 1} (\frac{a}{c}) \\ \tan B & = \frac{b}{a} B = \tan^{- 1} (\frac{b}{a}) \end{array}

Rule

Uses for Sine, Cosine, and Tangent

Finding the angles of a right triangle if you know two of the sides.
Finding the sides of a right triangle if you know one side and one angle.

Example 1

In a right triangle $△ A B C$ you know that the hypotenuse is $c = 10$ and one leg is $b = 8$ . You want to find the angle $∠ B$ , which is the angle opposite to $b$ .

It is useful to draw a figure to help you.

Example of using sine on a right triangle

Then you calculate

B = \sin^{- 1} (\frac{8}{10}) \approx {53.13}^{\circ}

Example 2

A right triangle $△ A B C$ has a leg $b = 3$ and angle $B = 30^{\circ}$ . Find the adjacent leg $a$ to the angle $∠ B$ .

Example of using tangent on a right triangle

As you know the angle $∠ B$ and the opposite leg to the angle $∠ B$ , it is natural to select the tangent function. You enter it into the formula and get

\begin{array}{l} \tan 30^{\circ} & = \frac{3}{a} & | \cdot a \\ a \cdot \tan 30^{\circ} & = 3 & | \div \tan 30^{\circ} \\ a & = \frac{3}{\tan 30^{\circ}} \approx 5.2 \end{array}

Example 3

Find the length of the hypotenuse $c$ in the right triangle $△ A B C$ when you know that $a = 3$ and that angle $∠ B = 60^{\circ}$ .

Example of using cosine on a right triangle

You need to find the hypotenuse, and you’ve been given the adjacent leg to angle $∠ B$ , so cosine is the best fit here. You input everything into the formula and get:

\begin{array}{l} \cos 60^{\circ} & = \frac{3}{c} & | \cdot c \\ c \cdot \cos 60^{\circ} & = 3 & | \div \cos 60^{\circ} \\ c & = \frac{3}{\cos 60^{\circ}} \\ c & = 6 \end{array}

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