9.4 The Tangent Ratio Opposite Side Adjacent Side Trigonometric Ratio Ratio of Opposite Side to Adjacent Side

Opposite and Adjacent Sides If you are “standing” at angle J identify the hypotenuse, the opposite side and the adjacent side. What if you are standing at angle K?

Since Trigonometric Ratios are constant for any given acute angle in a right triangle they can be tabulated into a Table of Trigonometric Ratios. Look at Tangent 52 ̊ 1.2799

In a right triangle with a 52̊ angle, if you “stand” at the 52̊ angle the ratio of the opposite side to the adjacent side will always be 1.2799 (from the Table of Trigonometric Ratios) Adj Opp

Calculating Trigonometric Ratios Use your calculator to find the value of the trigonometric ratio. Round to the nearest hundredth. tan 65° Be sure your calculator is in degree mode, not radian mode. Caution! tan 65°  2.14

Tangent Ratios for Special Right Triangles Find tan 45, tan 60 and tan 30. Leave your answers in reduced radical form: Step 1: Draw 45-45-90 and a 30-60-90 right triangles and place the ratios for the sides on the triangles. Step 2: Calculate the tangents

Angles of Elevation and Depression FROM Horizontal Down: Angle of Depression FROM Horizontal – Up: Angle of Elevation Since horizontal lines are parallel, 1  2 by the Alternate Interior Angles Theorem. Therefore the angle of elevation from one point is congruent to the angle of depression from the other point.

Check It Out! Example 5 What if…? Suppose a plane is at an altitude of 3500 ft and the angle of elevation from the airport to the plane is 29°. What is the horizontal distance between the plane and the airport? Round to the nearest foot.

Example 6: Finding Distance by Using Angle of Depression An ice climber stands at the edge of a crevasse that is 115 ft wide. The angle of depression from the edge where she stands to the bottom of the opposite side is 52º. How deep is the crevasse at this point? Round to the nearest foot.

Example 7: Shipping Application An observer in a lighthouse is 69 ft above the water. He sights two boats in the water directly in front of him. The angle of depression to the nearest boat is 48º. The angle of depression to the other boat is 22º. What is the distance between the two boats? Round to the nearest foot.

Check It Out! Example 8 A pilot flying at an altitude of 12,000 ft sights two airports directly in front of him. The angle of depression to one airport is 78°, and the angle of depression to the second airport is 19°. What is the distance between the two airports? Round to the nearest foot.

Check It Out! Example 9 A woman is standing 12 ft from a sculpture. The angle of elevation from her eye to the top of the sculpture is 30°, and the angle of depression to its base is 22°. How tall is the sculpture to the nearest foot?