A Quick Review ► We already know two methods for calculating unknown sides in triangles. ► We are now going to learn a 3 rd, that will also allow us to calculate unknown angles in right triangles. ► These methods are summarized in the chart below. Name of Method Triangle Type # of Sides Needed Math Involved Result 1. Pythagorean Theorem Right Angle 2a 2 +b 2 = c 2 Unknown side 2. Similar TrianglesAny 2 matching sides and 1 other side. Solving Ratios Unknown side 3. Trigonometry Right Angle ONLY! 2 sides or 1 side and 1 angle Solving Ratios Unknown side(s) and angle(s).

Trigonometry ► Trigonometry is the Greek word for Triangle Geometry. ► It uses three functions SINE, COSINE and TANGENT to explore how the sides and angles of a triangle are related. ► The TRIG RATIOS (sin, cos, tan) are ONLY used in the case of a RIGHT angle (90  ) triangle.

Hypotenuse Opposite Adjacent   Hypotenuse Opposite Adjacent Labeling the Sides of a Right Angle Triangle Always label according to the angle you are using.

The Tangent Ratio ► We will first explore a ratio called TANGENT. ► Think back to similar triangles… we set up ratios to solve for unknown sides. We are now going to use ratios to figure out unknown angles and sides. For all Trig ratios, calculators must be in degree mode: Mode – Degree

The ratio of the opposite side over the adjacent side in a right angle is called the TANGENT ratio. 7 5 A Tan Tan A = B Tan Tan B = 8.6 Tan Tan  4 A B C 3 C

Using TAN to solve for an unknown angle. ► Now that you know the ratio for TAN, we can use it to solve for unknown angles and sides. ► But first, let us look at the components of the ratio. Tan Tan  Side length Angle goes here (if it is known)

7 5 A Tan Tan A = B 8.6 Tan Tan  Calculator Sequence: 2nd function (or Inverse) TAN(5/7) What is the measure of Angle A? Tan Tan A = Does it make sense to leave your answer as 0.71? Why or Why not?

7 5 A B Tan Tan B = 8.6 Tan Tan  What is the measure of Angle B? How can we check to see if we have correctly calculated angle B?

Find the following ratios to four decimal places. c) tan 48° = e) tan 59° = a) tan 64° = b) tan 28° = d) tan 14° = f) tan 82° = What do these ratios mean?

27  Always write ratios to 4 decimal places. 5.0 Finding the length of an unknown side. In triangle ABC, <C=90 , <A=27 , and AC=5.0 cm. Calculate BC. A B C Tan Tan  Tan Tan 27  = = 5 x x = =

34  b A BC c 16 cm Finding the length of both unknown sides. Tan Tan  Step 1: Label your sides according to the angle in the question. Step 2: Which two sides are involved in the question?

A guy wire supports a tower. The wire forms an angle of 57  degrees with the level ground. The wire is attached to the ground 16.5 m from the base of the tower. 1.At what height is the guy wire attached to the tower? 2.How long is the guy wire?  = m Let’s try it! Tan Tan 

 Set up a Ratio using given sides:  D F H CEGI