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Solving Trigonometric Equations Using Inverses

Solving Trigonometric Equations Using Inverses . Section 14-2. 90°. 2 nd Quad. 1 st Quad. (- x , y ). ( x , y ). − cos θ + sin θ − tan θ. + cos θ + sin θ + tan θ. 180°. 0°. − cos θ − sin θ + tan θ. + cos θ − sin θ − tan θ. (- x , - y ). ( x , - y ).

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Solving Trigonometric Equations Using Inverses

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  1. Solving Trigonometric Equations Using Inverses Section 14-2

  2. 90° 2nd Quad. 1st Quad. (-x, y) (x, y) − cos θ + sin θ − tan θ + cos θ + sin θ + tan θ 180° 0° − cos θ − sin θ + tan θ + cos θ − sin θ − tan θ (-x, -y) (x, -y) 3rd Quad. 4th Quad. 270°

  3. 1. Open a calculator page on the N-Spire calculator. • 2. Press ctrl-home-file-document settings

  4. Press enter until your window looks like this.

  5. Today we are going to use these keys to solve trigonometric equations.

  6. Example 1 • Solve the following trigonometric equations in degrees. Round to nearest degree. • 1. sin θ = 0.31 • Rewrite the problem as sin-1 (0.31) = θ

  7. On your calculator press ctrl-sin and then 0.31. • Press enter. θ = 18°

  8. That is not the only solution. The sine ratio is also positive in the 2nd quadrant. To find the solution in the 2nd quadrant subtract your 1st solution from 180°. • θ = 180 – 18 • θ = 162° • Therefore, • θ = 18° or 162°

  9. 2. cos θ = 0.66 • θ = cos-1 (0.66) • θ = 49° • But cosine is also positive in the 4th quadrant. • To find the solution in the 4th quadrant subtract the 1st solution from 360°. • θ = 360 – 49 = 311° • Therefore • θ = 49° or 311°

  10. 3. tan θ = 4.12 • θ = tan-1 (4.12) • θ = 76° • But tangent is also positive in the 3rd quadrant. • To find the solution in the 3rd quadrant add the 1st solution to 180° • θ = 180 + 76 = 256° • Therefore, • θ = 76° or 256°

  11. Finding the 2nd Solution • 2nd Quadrant • Subtract the 1st solution from 180°. • 3rd Quadrant • Add the 1st solution to 180°. • 4th Quadrant • Subtract the 1st solution from 360°.

  12. Example 2 • Solve each equation in degrees to the nearest degree. • 1. 4 cos θ – 1 = cos θ • a. Add 1 to both sides. • 4 cos θ = cos θ + 1 • b. Subtract cos θ from both sides. • 3 cos θ = 1 • c. Divide both sides by 3.

  13. 2. 7 sin θ – 3 = 2 sin θ • 7 sin θ = 2 sin θ + 3 • 5 sin θ = 3

  14. 3. 4 tan θ = 3 + tan θ • 3 tan θ = 3 • tan θ = 1

  15. Example 3 • Solve the following trigonometric equations to the nearest degree. • 2 cos θ sin θ − sin θ = 0 • a. Since both terms on the left side have sin θ, factor it out of the left side. • sin θ(2 cos θ − 1) = 0

  16. b. Set each factor equal to 0. • sin θ = 0 2 cos θ − 1 = 0 • c. Solve each equation for θ. • sin θ = 0 • θ = sin-1 (0) • θ = 0° 1st solution • Sine is also equal to 0 when θ is equal to 180°. So θ = 180°

  17. 2 cos θ − 1 = 0 • 2 cos θ = 1

  18. The solutions for the problem are • θ = 0°, 180°, 60°, 300°

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