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Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.
Aim: How do we solve quadratic trigonometric equations?
Do Now:
Solve by factoring:x2 – 3x – 4 = 0
Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.
Solving Quadratic Equations
Solve by factoring: x2 – 3x – 4 = 0
x2 – 3x – 4 = 0(x – 4)(x + 1) = 0
(x – 4) = 0 (x + 1) = 0
x = 4 x = -1
Solve using quadratic formula: x2 – 3x – 4 = 0
x b b2 4ac
2a a = 1, b = -3, c = -4
x ( 3) ( 3)2 4(1)( 4)
2(1)
x 3 9 16
2
x 3 25
2x = 4 or -1
Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.
Quadratic Trig Equations - Factoring
tan2 – 3tan – 4 = 0
(tan – 4)(tan + 1) = 0
(tan – 4) = 0 (tan + 1) = 0
tan = 4 tan = -1
Solve for by factoring:to nearest degree, in the interval 0º ≤ ≤ 360º
tan is (–) in QII & QIV; reference = 45º
tan is (+) in QI & QIII; reference = 76º
QII 180 – 45 = 135º
QIV 360 – 45 = 315º
and
QI 76º
QIII 180 + 76 = 256º
and
{76º, 135º, 256º, 315º}
= 45º & 76º
Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.
Quadratic Trig Equations - Formula
tan2 – 3tan – 4 = 0to nearest degree, in the interval 0º ≤ ≤ 360º
tan is (–) in QII & QIV; reference = 45º
tan is (+) in QI & QIII; reference = 76º
QII 180 – 45 = 135º
QIV 360 – 45 = 315º
andQI 76º
QIII 180 + 76 = 256º
and
{76º, 135º, 256º, 315º}
tan b b2 4ac
2aa = 1, b = -3, c = -4
tan ( 3) ( 3)2 4(1)( 4)
2(1)
tan 3 9 16
2
tan 3 25
2tan = 4 and -1 = 45º & 76º
Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.
Model Problem
cos b b 2 4ac
2aa = 3, b = -5, c = -4
cos ( 5) ( 5)2 4(3)( 4)
2(3)
cos 5 73
62.25733
Given: 3cos2 – 5cos – 4 = 0, find to thenearest degree in the interval 0º ≤ ≤ 360º
cos 5 25 48
6
cos 5 73
6 .590667
Calculator
2nd 73 ENTER
Display: -.59066722909
( 5 –
) ) ÷ 6
2nd ENTER
Display: 53.7956245
(–) ANSCOS-1 2nd
= 54º
Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.
Model Problem (Con’t)
cos b b 2 4ac
2aa = 3, b = -5, c = -4
Given: 3cos2 – 5cos – 4 = 0, find to thenearest degree in the interval 0º ≤ ≤ 360º
= 54º
cosine is (–) in QII & QIII; reference = 54º
QII
180 + 54 = 234º
and
QIII
cos 5 73
6 .590667
180 – 54 = 126º
{126º, 234º}
Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.
Special Quadratics
Solve for in the interval 0º ≤ ≤ 360º: tan2 – 3 = 0
= 60º
tan2 = 3
tan 3
tan is (–) in QII & QIV; reference = 60º
tan is (+) in QI & QIII; reference = 60º
QII 180 – 60 = 120º
QIV 360 – 60 = 300º
andQI 60º
QIII 180 + 60 = 240º
and
{60º, 120º, 240º, 300º}
Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.
Model Problem
Solve the equation 2cos2 = cos , for all values of in the interval 0º ≤ ≤ 360º
standard form: 2cos2 – cos = 0cos(2cos – 1) = 0
cos = 0 (2cos – 1) = 0
factor & solve:
2cos = 1cos = 1/2 or .5
= 90º and 270º = 60º
cosine is (+) in QI & QIV; reference = 60º
QI 60º QIV 300 – 60 = 300ºand
{60º, 90º, 270º, 300º}
Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.
Model Problem
a = 2, b = -1, c = 02( 1) ( 1) 4(1)(0)
2(2)x
1 1 0
4x
2 0,
4 4x x = .5 and 0
Solve the equation 2cos2 = cos , for all values of in the interval 0º ≤ ≤ 360º
2cos2 – cos = 0
= 90º and 270º = 60º
cosine is (+) in QI & QIV; reference = 60º
QI 60º QIV 300 – 60 = 300ºand
{60º, 90º, 270º, 300º}
Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.
Regents Prep
Find all values of in the interval 0 < < 360o that satisfy the equation
2 sin2 + sin = 1.
Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.
Model Problem
a = 3, b = 4, c = -1
In the interval 0º ≤ ≤ 360º, find to the nearestdegree, for all values of that satisfy the equation
3sin 4 1
sinrewrite w/o fractions:
3sin2 4sin 1
standard form:
3sin2 4sin 1 0
sin b b2 4ac
2a
sin 4 42 4(3)( 1)
2(3)
sin 4 28
6
sin 4 28
6.215250437
sin 4 28
6 1.5485377
= 12.43º = 12º
Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.
Model Problem (Con’t)
In the interval 0º ≤ ≤ 360º, find to the nearestdegree, for all values of that satisfy the equation
3sin 4 1
sin
sin 4 28
6.215250437 = 12º
sine is (+) in QI & QII; reference = 12º
QI 12º QII 180 – 12 = 168ºand
{12º, 168º}
Aim: Solving Quadratic Trig Equations Course: Alg. 2 & Trig.