Presentation is loading. Please wait.

Presentation is loading. Please wait.

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

Similar presentations


Presentation on theme: "Trigonometric Equations In quadratic form, using identities or linear in sine and cosine."— Presentation transcript:

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

2 Solving a Trig Equation in Quadratic Form Solve the equation: 2sin 2 θ – 3 sin θ + 1 = 0, 0 θ 2 Let sin θ equal some variable sin θ = a Factor this equation (2a – 1) (a – 1) = 0 Therefore a = ½ a = 1

3 Solving a Trig Equation in Quadratic Form Now substitute sin θ back in for a sin θ = ½sin θ = 1 Now do the inverse sin to find what θ equals θ = sin -1 (½) θ = sin -1 1 θ = /6 and 5/6 θ = /2

4 Solving a Trig Equation in Quadratic Form Solve the equation: (tan θ – 1)(sec θ – 1) = 0 tan θ – 1 = 0sec θ – 1 = 0 tan θ = 1sec θ = 1 θ = tan -1 1 θ = sec -1 1 θ = /4 and 5/4 θ = 0

5 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 sin 2 θ Use the pythagorean identities to change sin 2 θ to cos θ

6 Solving a Trig Equation Using Identities sin 2 = 1 – cos 2 θ Substituting into the equation 3 cos θ + 3 = 2(1 – cos 2 θ) To solve a quadratic equation it must be equal to 0 2cos 2 θ + 3 cos θ + 1 = 0 Let cos θ = b

7 Solving a Trig Equation using Identities 2b 2 + 3b + 1 = 0 (2b + 1) (b + 1) = 0 (2b + 1) = 0b + 1 = 0 b = -½b = -1 cos θ = -½cos θ = -1 θ = 2/3, 4/3 θ =

8 Solving a Trig Equation Using Identities cos 2 θ – sin 2 θ + sin θ = 0 1 – sin 2 θ – sin 2 θ + sin θ = 0 -2sin 2 θ + sin θ + 1 = 0 2 sin 2 θ – sin θ – 1 = 0 Let c = sin θ 2c 2 – c – 1 = 0 (2c + 1) (c – 1) = 0

9 Solving a Trig Equation Using Identities (2c + 1) = 0c – 1 = 0 c = -½c = 1 sin θ = -½sin θ = 1 θ = + θ = θ = 4/3,

10 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 2sin 2 θ cos θ – cos θ = 0 cos θ(2sin 2 – 1) = 0 cos θ = 02sin 2 θ=1

11 Solving a Trig Equation Using Identities cos θ = 0 θ = 0, θ = /4,

12 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

13 Solving a Trig Equation Using Identities 2θ = 3 θ = 3/4 2θ = 3

14 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 = 1 2

15 Solving a Trig Equation Linear in sin θ and cos θ sin 2 θ + 2sin θ cos θ + cos 2 θ = 1 2sin θ cos θ + 1 = 1 2sin θ cos θ = 0 sin 2θ = 0 2θ = 2θ = θ = θ = θ = θ = 3

16 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

17 Solving a Trig Equation Linear in sin θ and cos θ Check: Does sin 0 + cos 0 = 1? Does sin cos Does sin 3cos 3

18 Solving a Trig Equation Linear in sin θ and cos θ sec θ = tan θ + cot θ sec 2 θ = (tan θ + cot θ) 2 sec 2 θ = tan 2 θ + 2 tan θ cot θ + cot 2 θ sec 2 θ = tan 2 θ cot 2 θ sec 2 θ – tan 2 θ = 2 + cot 2 θ 1 = 2 + cot 2 θ -1 = cot 2 θ

19 Solving a Trig Equation Linear in sin θ and cos θ is undefined (cant take the square root of a negative number).

20 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 y 1 Put 3 on y 2 Graph using the window 0 θ 2 Find the intersection point(s)

21 Word Problems Page 519 problem 58


Download ppt "Trigonometric Equations In quadratic form, using identities or linear in sine and cosine."

Similar presentations


Ads by Google