Lab 6.5 Trigonometric Equations PDF
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This document contains a set of trigonometric equations. It includes problems related to solving trigonometric equations and finding exact values of trigonometric expressions. The problems utilize trigonometric identities which involve sum and difference identities, double/half angle identities.
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Lab 6.5 Trigonometric Equations Solve the following equation. Give all solutions in radians and within the interval [0, 2π). 2 1. 2𝑐𝑜𝑠 𝑥 + 𝑠𝑖𝑛 𝑥 − 1 = 0 2. 𝑠𝑖𝑛 (2𝑥) = 𝑐𝑜𝑠 𝑥 3. (𝑡𝑎𝑛 𝑥 − 1)(𝑐𝑜𝑠 𝑥 − 1) = 0 4. (2𝑐𝑜𝑠 𝑥 − 3 )(2𝑠𝑖𝑛 𝑥 − 1) = 0 5. 𝑐𝑜𝑠 (2𝑥) = 𝑐𝑜𝑠 𝑥...
Lab 6.5 Trigonometric Equations Solve the following equation. Give all solutions in radians and within the interval [0, 2π). 2 1. 2𝑐𝑜𝑠 𝑥 + 𝑠𝑖𝑛 𝑥 − 1 = 0 2. 𝑠𝑖𝑛 (2𝑥) = 𝑐𝑜𝑠 𝑥 3. (𝑡𝑎𝑛 𝑥 − 1)(𝑐𝑜𝑠 𝑥 − 1) = 0 4. (2𝑐𝑜𝑠 𝑥 − 3 )(2𝑠𝑖𝑛 𝑥 − 1) = 0 5. 𝑐𝑜𝑠 (2𝑥) = 𝑐𝑜𝑠 𝑥 2 6. 4𝑠𝑖𝑛 𝑥 + 4𝑐𝑜𝑠 𝑥 − 5 = 0 2 7. 2𝑡𝑎𝑛 𝑥 + 5𝑡𝑎𝑛 𝑥 + 3 = 0 8. 𝑐𝑜𝑠 𝑥 − 5 = 3𝑐𝑜𝑠 𝑥 + 6 Solve the following equation using a calculator. Give all the solutions in radians rounded to 2 decimal places and within the interval [ 0, 2𝜋 ) 9. 𝑠𝑖𝑛 𝑥 = 0. 8246 10. 𝑐𝑜𝑠 𝑥 =− 0. 4721 11. 𝑠𝑖𝑛 𝑥 = 1. 1414 12. 𝑡𝑎𝑛 𝑥 =− 1. 1285 Simplify into a single Trig Function 𝑐𝑜𝑠 (𝑥) 𝑠𝑒𝑐 (𝑥) 13. 𝑡𝑎𝑛 (𝑥) 𝑡𝑎𝑛 (𝑥) − 𝑐𝑜𝑡 (−𝑥) 14. 𝑠𝑒𝑐 (𝑥) 2 (𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥) −1 15. 2 𝑐𝑜𝑠 𝑥 1 16. (𝑡𝑎𝑛 𝑦 + 𝑐𝑜𝑡 𝑦)𝑠𝑖𝑛 𝑦 𝑐𝑠𝑐 𝑥 − 𝑐𝑜𝑡 𝑥 17. 𝑠𝑒𝑐 𝑥 −1 𝑐𝑜𝑠 𝑥 18. 1−𝑠𝑖𝑛 𝑥 + tan (−x) Use a double-angle or half-angle identity to find the exact value of each expression. 7 θ 19. 𝑠𝑖𝑛 θ = − and 270° < θ < 360°. Find 𝑐𝑜𝑠 25 2 4 20. 𝑐𝑜𝑠 θ = and 270° < θ < 360°. Find 𝑠𝑖𝑛 2θ 5 3 3π θ 21. 𝑠𝑖𝑛 θ = − and < θ < 2π. Find 𝑡𝑎𝑛 5 2 2 3 91 3π 22. 𝑐𝑜𝑡 θ = − and < θ < 2π. Find 𝑡𝑎𝑛 2θ 91 2 13 π 23. 𝑠𝑒𝑐 θ = − 5 and 2 < θ < π. Find 𝑐𝑜𝑠 2θ Use a sum and difference identity to find the exact value of each expression. 24. 𝑠𝑖𝑛 (− 15°) 25. 𝑐𝑜𝑠 (− 105°) 5π 26. 𝑡𝑎𝑛 12 π 2π π 2π 27. 𝑠𝑖𝑛 9 𝑐𝑜𝑠 9 + 𝑐𝑜𝑠 9 𝑠𝑖𝑛 9 𝑡𝑎𝑛 76° + 𝑡𝑎𝑛 164° 28. 1 − 𝑡𝑎𝑛 76° 𝑡𝑎𝑛 164°