Trigonometric Equations

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# Trigonometric Equations - PowerPoint PPT Presentation

Trigonometric Equations. In quadratic form, using identities or linear in sine and cosine. Solving a Trig Equation in Quadratic Form. Solve the equation: 2sin 2 θ – 3 sin θ + 1 = 0, 0 ≤ θ ≤ 2 p Let sin θ equal some variable sin θ = a Factor this equation (2a – 1) (a – 1) = 0

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### Trigonometric Equations

In quadratic form, using identities or linear in sine and cosine

Solving a Trig Equation in Quadratic Form
• Solve the equation:
• 2sin2θ – 3 sin θ + 1 = 0, 0 ≤ θ≤ 2p
• Let sin θ equal some variable
• sin θ = a
• Factor this equation
• (2a – 1) (a – 1) = 0
• Therefore a = ½ a = 1
Solving a Trig Equation in Quadratic Form
• Now substitute sin θ back in for a
• sin θ = ½ sin θ = 1
• Now do the inverse sin to find what θ equals
• θ = sin-1 (½) θ = sin-1 1
• θ = p/6 and 5p/6 θ = p/2
Solving a Trig Equation in Quadratic Form
• Solve the equation:
• (tan θ – 1)(sec θ – 1) = 0
• tan θ – 1 = 0 sec θ – 1 = 0
• tan θ = 1 sec θ = 1
• θ = tan-1 1 θ = sec-1 1
• θ = p/4 and 5p/4 θ = 0
Solving a Trig Equation Using Identities
• In order to solve trig equations, we want to have a single trig word in the equation. We can use trig identities to accomplish this goal.
• Solve the equation
• 3 cos θ + 3 = 2 sin2θ
• Use the pythagorean identities to change sin2 θ to cos θ
Solving a Trig Equation Using Identities
• sin2 = 1 – cos2 θ
• Substituting into the equation
• 3 cos θ + 3 = 2(1 – cos2θ)
• To solve a quadratic equation it must be equal to 0
• 2cos2θ + 3 cos θ + 1 = 0
• Let cos θ = b
Solving a Trig Equation using Identities
• 2b2 + 3b + 1 = 0
• (2b + 1) (b + 1) = 0
• (2b + 1) = 0 b + 1 = 0
• b = -½ b = -1
• cos θ = -½ cos θ = -1
• θ = 2p/3, 4p/3 θ = p
Solving a Trig Equation Using Identities
• cos2θ – sin2θ + sin θ = 0
• 1 – sin2θ – sin2θ + sin θ = 0
• -2sin2 θ + sin θ + 1 = 0
• 2 sin2θ – sin θ – 1 = 0
• Let c = sin θ
• 2c2 – c – 1 = 0
• (2c + 1) (c – 1) = 0
Solving a Trig Equation Using Identities
• (2c + 1) = 0 c – 1 = 0
• c = -½ c = 1
• sin θ = -½ sin θ = 1
• θ = p/3 + p q=2p-p/3θ = p/2
• θ = 4p/3, q = 7p/3
Solving a Trig Equation Using Identities
• Solve the equation
• sin (2θ) sin θ = cos θ
• Substitute in the formula for sin 2θ
• (2sin θ cos θ)sin θ=cos θ
• 2sin2 θ cos θ – cos θ = 0
• cos θ(2sin2 – 1) = 0
• cos θ = 0 2sin2 θ=1
Solving a Trig Equation Using Identities
• cos θ = 0
• θ = 0, pθ = p/4, 3p/4, 5p/4, 7p/4
Solving a Trig Equation Using Identities
• sin θ cos θ = -½
• This looks very much like the sin double angle formula. The only thing missing is the two in front of it.
• So . . . multiply both sides by 2
• 2 sin θ cos θ = -1
• sin 2θ = -1
• 2 θ = sin-1 -1
Solving a Trig Equation Using Identities
• 2θ = 3p/2
• θ = 3p/4 2θ = 3p/2 + 2p

2q = 7p/2

q = 7p/4

Solving a Trig Equation Linear in sin θ and cos θ
• sin θ + cos θ = 1
• There is nothing I can substitute in for in this problem. The best way to solve this equation is to force a pythagorean identity by squaring both sides.
• (sin θ + cos θ)2 = 12
Solving a Trig Equation Linear in sin θ and cos θ
• sin2θ + 2sin θ cos θ + cos2 θ = 1
• 2sin θ cos θ + 1 = 1
• 2sin θ cos θ = 0
• sin 2θ = 0
• 2θ = 0 2θ = p
• θ = 0 θ = p/2
• θ = p θ = 3p/2
Solving a Trig Equation Linear in sin θ and cos θ
• Since we squared both sides, these answers may not all be correct (when you square a negative number it becomes positive).
• In the original equation, there were no terms that were squared
Solving a Trig Equation Linear in sin θ and cos θ
• Check:
• Does sin 0 + cos 0 = 1?
• Does sin p/2 + cos p/2 = 1?
• Does sin p + cos p = 1?
• Does sin 3p/2 + cos 3p/2 = 1?
Solving a Trig Equation Linear in sin θ and cos θ
• sec θ = tan θ + cot θ
• sec2 θ = (tan θ + cot θ)2
• sec2 θ = tan2 θ + 2 tan θ cot θ + cot2 θ
• sec2 θ = tan2 θ + 2 + cot2 θ
• sec2 θ – tan2 θ = 2 + cot2 θ
• 1 = 2 + cot2 θ
• -1 = cot2 θ
Solving a Trig Equation Linear in sin θ and cos θ
• q is undefined (can’t take the square root of a negative number).
Solving Trig Equations Using a Graphing Utility
• Solve 5 sin x + x = 3. Express the solution(s) rounded to two decimal places.
• Put 5 sin x + x on y1
• Put 3 on y2
• Graph using the window 0 ≤ θ ≤2p
• Find the intersection point(s)
Word Problems
• Page 519 problem 58