Trigonometric Equations

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
• 2sin2 ? 3 sin ? 1 0, 0 ? 2p
• Let sin ? equal some variable
• sin ? 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 ? back in for a
• sin ? ½ sin ? 1
• Now do the inverse sin to find what ? equals
• ? sin-1 (½) ? sin-1 1
• ? p/6 and 5p/6 ? p/2

4
Solving a Trig Equation in Quadratic Form

• Solve the equation
• (tan ? 1)(sec ? 1) 0
• tan ? 1 0 sec ? 1 0
• tan ? 1 sec ? 1
• ? tan-1 1 ? sec-1 1
• ? p/4 and 5p/4 ? 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 ? 3 2 sin2 ?
• Use the pythagorean identities to change sin2 ?
  to cos ?

6
Solving a Trig Equation Using Identities

• sin2 1 cos2 ?
• Substituting into the equation
• 3 cos ? 3 2(1 cos2 ?)
• To solve a quadratic equation it must be equal to
  0
• 2cos2 ? 3 cos ? 1 0
• Let cos ? b

7
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 ? -½ cos ? -1
• ? 2p/3, 4p/3 ? p

8
Solving a Trig Equation Using Identities

• cos2 ? sin2 ? sin ? 0
• 1 sin2 ? sin2 ? sin ? 0
• -2sin2 ? sin ? 1 0
• 2 sin2 ? sin ? 1 0
• Let c sin ?
• 2c2 c 1 0
• (2c 1) (c 1) 0

9
Solving a Trig Equation Using Identities

• (2c 1) 0 c 1 0
• c -½ c 1
• sin ? -½ sin ? 1
• ? p/3 p q2p-p/3 ? p/2
• ? 4p/3, q 7p/3

10
Solving a Trig Equation Using Identities

• Solve the equation
• sin (2?) sin ? cos ?
• Substitute in the formula for sin 2?
• (2sin ? cos ?)sin ?cos ?
• 2sin2 ? cos ? cos ? 0
• cos ?(2sin2 1) 0
• cos ? 0 2sin2 ?1

11
Solving a Trig Equation Using Identities

• cos ? 0
• ? 0, p ? p/4, 3p/4, 5p/4, 7p/4

12
Solving a Trig Equation Using Identities

• sin ? cos ? -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 ? cos ? -1
• sin 2? -1
• 2 ? sin-1 -1

13
Solving a Trig Equation Using Identities

• 2? 3p/2
• ? 3p/4 2? 3p/2 2p
• 2q 7p/2
• q 7p/4

14
Solving a Trig Equation Linear in sin ? and cos ?

• sin ? cos ? 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 ? cos ?)2 12

15
Solving a Trig Equation Linear in sin ? and cos ?

• sin2 ? 2sin ? cos ? cos2 ? 1
• 2sin ? cos ? 1 1
• 2sin ? cos ? 0
• sin 2? 0
• 2? 0 2? p
• ? 0 ? p/2
• ? p ? 3p/2

16
Solving a Trig Equation Linear in sin ? and cos ?

• 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 ? and cos ?

• 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?

18
Solving a Trig Equation Linear in sin ? and cos ?

• sec ? tan ? cot ?
• sec2 ? (tan ? cot ?)2
• sec2 ? tan2 ? 2 tan ? cot ? cot2 ?
• sec2 ? tan2 ? 2 cot2 ?
• sec2 ? tan2 ? 2 cot2 ?
• 1 2 cot2 ?
• -1 cot2 ?

19
Solving a Trig Equation Linear in sin ? and cos ?

• q 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 y1
• Put 3 on y2
• Graph using the window 0 ? 2p
• Find the intersection point(s)

21
Word Problems

• Page 519 problem 58