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Pre-Calculus

Pre-Calculus. Chapter 4 Trigonometric Functions. 4.3 Right Triangle Trigonometry. Objectives: Evaluate trigonometric functions of acute angles. Use fundamental trigonometric identities. Use a calculator to evaluate trigonometric functions.

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Pre-Calculus

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  1. Pre-Calculus Chapter 4 Trigonometric Functions

  2. 4.3 Right Triangle Trigonometry Objectives: • Evaluate trigonometric functions of acute angles. • Use fundamental trigonometric identities. • Use a calculator to evaluate trigonometric functions. • Use trigonometric functions to model and solve real-life problems.

  3. Right Triangle Definitions of Trigonometric Functions • Write the six trigonometric functions of angle θ using the right triangle shown below. • Note that θ is an acute angle. That is, 0°≤ θ≤ 90°. (θ lies in the first quadrant.)

  4. Example 1 • Find the exact values of the six trig functions of the angle θ shown in the figure.

  5. Trig Functions of Special Angles • Use special triangles to find the exact values of the trig functions of angles 45°, 30°, and 60°. 30° 45° 60°

  6. Summary of Special Angles Note: • sin 30° = cos60° • sin 60° = cos30° • sin 45° = cos45°

  7. Cofunctions • Cofunctions of complementary angles are equal. • sin (90° - θ) = cosθcos(90° - θ) = sin θ • tan (90° - θ) = cot θ cot (90° - θ) = tan θ • sec (90° - θ) = cscθcsc(90° - θ) = sec θ • Cofunctions are sine & cosine, tangent & cotangent, secant & cosecant.

  8. Fundamental Trig Identities • List the six reciprocal identities. • List the two quotient identities. • List the three Pythagorean identities.

  9. Example 2 • Let be θan acute angle such that sin θ = 0.6. Use trig identities to find: • cosθ • tan θ

  10. Example 3 • Use trig identities to transform one side of the equation into the other (0 < θ < π/2). • cosθ sec θ = 1 • (sec θ + tan θ)(sec θ – tan θ) = 1

  11. Angles of Elevation and Depression • Angle of Elevation The angle from the horizontal upward to the object. • Angle of Depression The angle from the horizontal downward to the object.

  12. Example 4 • A surveyor is standing 50 feet from the base of a large tree, as shown in the figure. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree?

  13. Example 5 • Find the length c of the skateboard ramp shown in the figure. 18.4°

  14. Example 6 • In traveling across flat land you notice a mountain directly in front of you. Its angle of elevation (to the peak) is 3.5°. After you drive 13 miles closer to the mountain, the angle of elevation is 9°. Approximate the height of the mountain.

  15. Homework 4.3 • Worksheet 4.3

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