Solving trigonometric equations 2
- 2. Objectives o Solving Trigonometric Equations. o Find extraneous solutions from trigonometric equations.
- 5. Remember that solving a trigonometric equations means solving for all values of the variable. Solve 2 𝑠𝑒𝑐2 ∅ − 𝑡𝑎𝑛4 ∅ = −1,for all values of ∅ if ∅ is measured by degrees. Solution: *use 𝑠𝑒𝑐2 ∅ = 1 + 𝑡𝑎𝑛2 ∅ 2 1 + 𝑡𝑎𝑛2 ∅ − 𝑡𝑎𝑛4 ∅ = −1 * Use distributive property: 2 + 2𝑡𝑎𝑛2 ∅ − 𝑡𝑎𝑛4 ∅ = −1 *Set one side of the equation equal to 0 𝑡𝑎𝑛4 ∅ − 2𝑡𝑎𝑛2 ∅ − 3 = 0 *Factor 𝑡𝑎𝑛2 ∅ − 3 𝑡𝑎𝑛2 ∅ + 1 = 0 *Use product property: 𝑡𝑎𝑛2∅ − 3 =0 or ( 𝑡𝑎𝑛2∅ + 1) = 0 𝑡𝑎𝑛2 ∅ = 3 𝑡𝑎𝑛2 ∅ = −1 (𝑡ℎ𝑖𝑠 𝑝𝑎𝑟𝑡 𝑔𝑖𝑣𝑒𝑠 𝑛𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑠𝑖𝑛𝑐𝑒 𝑡𝑎𝑛2 ∅ 𝑖𝑠 𝑛𝑒𝑣𝑒𝑟 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 ) 𝑡𝑎𝑛2∅ = ∓ 3 ∅ = 60° + 180°𝑘 and ∅ = −60° + 180°𝑘, where 𝑘 is any integer Solving Trigonometric Equations by using Identities
- 6. Solve the equation sin ∅ cot ∅ − cos2∅ = 0 Solution: *cot ∅ = cos∅ sin∅ sin ∅ cos∅ sin∅ − cos2∅ = 0 cos ∅ − cos2 ∅ = 0 *Factor cos ∅ cos ∅ (1 − cos ∅) = 0 *Use product property: cos ∅ = 0 or 1 − cos ∅ = 0 ∅ = π 2 + 2k or cos ∅ = 1 Then, cos ∅ = 1 cos−1 1 = ∅ ∅ = 0 Solving Trigonometric Equations by using Identities
- 7. Solving Trigonometric Equations by using Identities practice Solve 9 + 𝑐𝑜𝑡2 𝑥 = 12 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑥.
- 8. extraneous solutions Some trigonometric equations have no solution. For example, the equation 𝑐𝑜𝑠∅ = 4 has no solution because all values of 𝑐𝑜𝑠∅ are between −1 𝑎𝑛𝑑 1. Thus, the solution of 𝑐𝑜𝑠∅ = 4 is empty.
- 9. When you ride a Ferris wheel that has a diameter of 40 meters and turn at a rate of 1.5 revolutions per minute, the height above the ground in meters of your seat after 𝑡 minutes can be by the equation ℎ = 21 − 20 cos 3𝜋𝑡. After the ride begins, how long is it before your seat is 31 meters above the ground for the first time? Solution: *Replace ℎ with 31. 31 = 21 − 20 cos 3𝜋𝑡 *subtract 21 from each sides 10 = −20 cos 3𝜋𝑡 *take the arccosine cos−1 − 1 2 = 3𝜋𝑡 The arccosine of − 1 2 𝑖𝑠 2𝜋 3 𝑜𝑟 4𝜋 3 So, 2𝜋 3 = 3𝜋𝑡 or 4𝜋 3 = 3𝜋𝑡 2𝜋 3 + 2𝜋𝑘 = 3𝜋𝑡 or 4𝜋 3 + 2𝜋𝑘 = 3𝜋𝑡 * Divide each term by 3𝜋 2 9 + 2 3 𝑘 = 𝑡 or 4 9 + 2 3 𝑘 = 𝑡 The least positive value for 𝑡 is obtained by letting 𝑘 = 0 in the first expression. Therefore, 𝑡 = 2 9 of a minute or 13 seconds. Solve trigonometric equations
- 10. Solve trigonometric equations practice The tallest building in the world Burj Khalifa in Dubai, UAE, with 2,717 feet. What is the measure of ∅ if the length of its shadow is 1 mile?