8 3 8 4 Sine Cosine and Tangent

8 -3 8 -4 Sine, Cosine and Tangent Ratios Objective Students will be able to use sine, cosine, and tangent ratios to determine side lengths in triangles.

Trigonometric Ratios • We use the Pythagorean Theorem when we are given two sides of a right triangle and we want to find the third. • We will use trigonometric ratios when we are given one side and one angle (other than the 90 degree angle) of a right triangle and want to find another (or both) side(s). • There are three trigonometric ratios we will utilize • Sine (abbreviated sin) • Cosine (abbreviated cos) • Tangent (abbreviated tan)

Side Definitions • Opposite side: leg directly across from the angle of interest • Adjacent side: leg next to the angle of interest • Hypotenuse: side directly across from the right angle • The opposite and adjacent sides differ depending on the angle of interest. For example, if you are looking at angle X, then the opposite side is a. However, if you are looking at angle Y, then the opposite side is b.

Other notes • Never use the right angle when using trigonometric ratios. Only use one of the two acute angles. • Calculators must be in “degree” mode. To check, press the “mode” button and go down to where you see “radian” and “degree”. If not already highlighted, highlight “degree” and press enter. • If your calculator is in “radian” mode, you will not get the correct answers we are looking for here. (They are correct answers but for Geometry we want answers in Degree Mode)

Example 1: Write the sine, cosine, and tangent ratios for angles T and U.

You Try Write the sine, cosine, and tangent ratios for angles J and K.

Finding Side Length • When using trigonometry to solve for a side length, first determine which trig ratio to use based on the given information. • Then, substitute in the information. • Finally, solve as you would solve a proportion. We usually round to the nearest tenth.

Example 2: Find the value of x to the nearest tenth. 1) 2)

Example 2 b: Find the value of x to the nearest tenth. 3) 4)

Try these 5) 7) 6)

What if my variable is in the Denominator? 8) 9)

Fun Stuff! 10) 11)

Just some more… 12) 13)

Finding Angle Measures • If given two sides of a right triangle, we can determine the angle measures by using inverse trigonometric ratios. • Start by determining the appropriate ratio to use and substituting in your information. • Then, take the inverse of the ratio. • To do this on the calculator, hit “ 2 nd” and then hit either sin, cos, or tan (depending on which ratio is appropriate given the problem).

Ex: Finding Angle Measure

Examples: Finding Angles Find the value of x. Round answers to the nearest degree. 1) 2)

Examples: Finding Angles 3) 4)

Try these… 5) 6)

Critical Thinking • Describe and create a triangle where:

Critical Thinking • Given the right triangle ABC where C is the Right Angle, determine if the following statement is valid. Explain why or why not.