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Exploring Trigonometric Ratios in Right Triangles

Learn about sine, cosine, and tangent ratios in right triangles with practical examples and applications. Understand angle of elevations, depressions, and how to solve trigonometric equations.

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Exploring Trigonometric Ratios in Right Triangles

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  1. Chapter 13 Sec 1 Right Triangle Trigonometry

  2. Trigonometric Ratios • The ratios of the sides of the right triangle can be used to define the trigonometric ratios. • The ratio of the side opposite θand the hypotenuse is known as sine. • The ratio of the side adjacent θ and the hypotenuse is known as cosine. • The ratio of the side opposite θ and the side adjacent θ is known as tangent.

  3. Right Triangle Trigonometry • Let’s consider a right triangle, one of whose acute angles is θ • The three sides of the triangle are the hypotenuse, the side oppositeθ, and the side adjacent to θ . opposite hypotenuse SOH CAH TOA adjacent

  4. Example 1 Find the values of the sine, cosine, and tangent for A. First find the length of AC. (AB)2 + (BC)2 = (AC)2 152 + 82 = 289 = (AC)2 AC = 17 C 8 cm B 15 cm A 17 cm

  5. Special Values

  6. Example 2 Write an equation involving sin, cos, or tan that could be used to find the value of x. Then solve the equation. Round to the nearest tenth. 8 30° x

  7. Example 3 Solve ∆XYZ. Round measures of the sides to the nearest tenth and measures of angles to the nearest degree. X 10 35° z ZxY Find x and y Find Y

  8. Example 4 Solve ∆ABC. Round measures of the sides to the nearest tenth and measures of angles to the nearest degree. B 13 5 C 12 A Find A Use a calculator and the SIN–1 function to find the angle whose sine is 5/13 . Find B

  9. Example 5 In order to construct a bridge across a river, the width of the river at the location must be determined. Suppose a stake is planted on one side of the river directly across from a second stake on the opposite side. At a distance 30 meters to the right of the stake, an angle of 55°, find the width of the river. w 55° 30 m

  10. Elevation and Depression • There are many applications requiring trigonometric solutions. A prime example would be surveyors use of special instruments to find the measures of angles of elevation and angles of depression. • Angle of elevations is the angle between a horizontal line and the line of sight from an observer to an object at a higher level. • Angle of depression is the angle between a horizontal line and the line of sight from the observer to an object at a lower level. • These two are equal measures because they are alternate interior angles.

  11. Example 6 The Aerial run in Snowbird, Utah, has an angle of elevation of 20.2°. It’s vertical drop is 2900 feet. Estimate the length of this run.

  12. Daily Assignment • Chapter 13 Section 1 • Study Guide • Pg 175 – 176 All

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