Question

Solve the following trigonometric equations on the interval 0 ? ? ? 2?. Express your answer in radians. If a radian angle is not a standard angle, you may round correctly to two decimal places. You will get two of the following, chosen at random: (a) (sin^2(?) - 9)(2 cos(?) + 1) = 0 (b) csc(?) = -5 (c) tan(?) = -10 2. () Solve the following trigonometric equations on the interval -? ? ? ? ?. Express your answer in radians. If a radian angle is not a standard angle, you may round correctly to two decimal places. You will get two of the following, chosen at random: (a) sin(?) = -0.45 (b) cos(?) sin(?) + 4 cos(?) = 0 (c) cos^2(?) + 2 cos(?) - 3 = 0 3. () Find all solutions of the following trigonometric equations by first finding two angles that solve the equation on the interval [0, 2?]. Express your answer in radians. If a radian angle is not a standard angle, you may round correctly to two decimal places. You will get two of the following, chosen at random: (a) cos(3?) = 0.75 (b) 1 - 2 sin(5?) = 0 (c) 2 tan(?) + sec^2(?) = 4

          Solve the following trigonometric equations on the interval 0 ? ? ? 2?. Express your answer in radians. If a radian angle is not a standard angle, you may round correctly to two decimal places. You will get two of the following, chosen at random:
(a) (sin^2(?) - 9)(2 cos(?) + 1) = 0
(b) csc(?) = -5
(c) tan(?) = -10

2. (*) Solve the following trigonometric equations on the interval -? ? ? ? ?. Express your answer in radians. If a radian angle is not a standard angle, you may round correctly to two decimal places. You will get two of the following, chosen at random:
(a) sin(?) = -0.45
(b) cos(?) sin(?) + 4 cos(?) = 0
(c) cos^2(?) + 2 cos(?) - 3 = 0

3. (*) Find all solutions of the following trigonometric equations by first finding two angles that solve the equation on the interval [0, 2?]. Express your answer in radians. If a radian angle is not a standard angle, you may round correctly to two decimal places. You will get two of the following, chosen at random:
(a) cos(3?) = 0.75
(b) 1 - 2 sin(5?) = 0
(c) 2 tan(?) + sec^2(?) = 4

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Solve the following trigonometric equations on the interval 0 ? ? ? 2?. Express your answer in radians. If a radian angle is not a standard angle, you may round correctly to two decimal places. You will get two of the following, chosen at random:
(a) (sin^2(?) - 9)(2 cos(?) + 1) = 0
(b) csc(?) = -5
(c) tan(?) = -10

2. (*) Solve the following trigonometric equations on the interval -? ? ? ? ?. Express your answer in radians. If a radian angle is not a standard angle, you may round correctly to two decimal places. You will get two of the following, chosen at random:
(a) sin(?) = -0.45
(b) cos(?) sin(?) + 4 cos(?) = 0
(c) cos^2(?) + 2 cos(?) - 3 = 0

3. (*) Find all solutions of the following trigonometric equations by first finding two angles that solve the equation on the interval [0, 2?]. Express your answer in radians. If a radian angle is not a standard angle, you may round correctly to two decimal places. You will get two of the following, chosen at random:
(a) cos(3?) = 0.75
(b) 1 - 2 sin(5?) = 0
(c) 2 tan(?) + sec^2(?) = 4

Added by Jane R.

Precalculus with Limits

Ron Larson 2nd Edition

Instant Answer

Solved by Expert Julian Wong

08/23/2022

Step 1

1.1: sinθ - 9 = 0 sinθ = 9 This has no solution as sinθ cannot be greater than 1. 1.2: 2cosθ + 1 = 0 2cosθ = -1 cosθ = -1/2 θ = 2π/3 ** Show more…

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Solve the following trigonometric equations on the interval 0 ≤ θ ≤ 2π. Express your answer in radians. If a radian angle is not a standard angle, you may round correctly to two decimal places. You will get two of the following, chosen at random: (a) (sin^2(θ) - 9)(2 cos(θ) + 1) = 0 (b) csc(θ) = -5 (c) tan(θ) = -10 Solve the following trigonometric equations on the interval -π ≤ θ ≤ π. Express your answer in radians. If a radian angle is not a standard angle, you may round correctly to two decimal places. You will get two of the following, chosen at random: (a) sin(θ) = -0.45 (b) cos(θ) sin(θ) + 4 cos(θ) = 0 (c) cos^2(θ) + 2 cos(θ) - 3 = 0 Find all solutions of the following trigonometric equations by first finding two angles that solve the equation on the interval [0, 2π]. Express your answer in radians. If a radian angle is not a standard angle, you may round correctly to two decimal places. You will get two of the following, chosen at random: (a) cos(3θ) = 0.75 (b) 1 - 2 sin(5θ) = 0 (c) 2 tan(θ) + sec^2(θ) = 4

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Transcript

00:01 So for 1a, we can simply take both terms and set them equal to 0 in order to find the values of theta that make the equation equals 0 or the statement to be true.

00:19 So set this equal to 0 first, add 9 to both sides, and we get sine squared of theta is equal to 9.

00:29 Theta here doesn't have a solution because if you take the square root of both sides, it would be sine of theta is equal to 3, and since 3 is greater than 1, there is no solution.

00:43 However, on the other side, if you move one to the other side and divide by 2 from both sides, we get that the cosine of theta is equal to 1⁄2, and that's a pretty simple trigonometric relationship.

01:00 So, theta will be equal to pi over 3.

01:06 And since that's the only solution of the two, this is the only solution.

01:13 So 1b states that sine of theta is equal to, or 1 over the sign of theta is equal to negative 5.

01:20 If you take the reciprocal of both sides, we get that sine of theta is equal to negative 1 over 5.

01:28 There's nothing you can do but plug this into a calculator.

01:31 Theta is equal to the arc sine of negative 1 over 5.

01:33 Theta is equal to 3 .34.

01:36 So for 1c, we get that the tangent of theta is equal to negative 10.

01:41 Again, we can't do anything but plug in.

01:43 Theta is equal to the arc tangent or the inverse tangent of negative 10.

01:48 And once you do that, theta is equal to 1 .67.

01:52 So for 2a, we get that sine of theta is equal to negative 0 .45.

01:57 Again, we can't do anything but plug this into a calculator.

02:00 Theta is equal to the arc sign of negative 0 .45.

02:04 That is equal to 3 .61.

02:07 So for 2b, we have a long expanded polynomial of trigonometric expressions.

02:19 If you factor out cosine of theta, which i already did here, you get this expression right here.

02:26 And again, this is two terms multiplied together to equal to zero, so we can set both equal to zero and then solve for theta.

02:32 So sine of theta is equal to four.

02:34 That has no solution because 4 is greater than 1.

02:36 Cosine of theta is equal to 0...

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