CHAPTER 10 Geometry

Right Triangle Trigonometry 10.6 Right Triangle Trigonometry

Objectives Use the lengths of the sides of a right triangle to find trigonometric ratios. Use trigonometric ratios to find missing parts of right triangles. Use trigonometric ratios to solve applied problems.

Ratios in Right Triangles Trigonometry means measurement of triangles. Trigonometric Ratios: Let A represent an acute angle of a right triangle, with right angle, C, shown here.

Ratios in Right Triangles For angle A, the trigonometric ratios are defined as follows:

Example: Becoming Familiar with The Trigonometric Ratios Find the sine, cosine, and tangent of A. Solution: Using the Pythagorean Theorem, find the measure of the hypotenuse c.

Example: Finding a Missing Leg of a Right Triangle Find a in the right triangle Solution: Because we have a known angle, 40°, with a known tangent ratio, and an unknown opposite side, “a,” and a known adjacent side, 150 cm, we can use the tangent ratio. tan 40° = a = 150 tan 40° ≈ 126 cm

Applications of the Trigonometric Ratios Angle of elevation: Angle formed by a horizontal line and the line of sight to an object that is above the horizontal line. Angle of depression: Angle formed by a horizontal line and the line of sight to an object that is below the horizontal line.

Example: Problem Solving using an Angle of Elevation Find the approximate height of this tower. Solution: We have a right triangle with a known angle, 57.2°, an unknown opposite side, and a known adjacent side, 125 ft. Using the tangent ratio: tan 57.2° = a = 125 tan 57.2° ≈ 194 feet

Example: Determining the Angle of Elevation A building that is 21 meters tall casts a shadow 25 meters long. Find the angle of elevation of the sun. Solution: We are asked to find mA.

Example continued Use the inverse tangent key The display should show approximately 40. Thus the angle of elevation of the sun is approximately 40°.