Mathematics, 20.07.2019 20:30 bellsimps7179

What are the values that satisfy the trigonometric equation for: 0 ≤ q ≤ 2π ? sinθ + tan(-θ) = 0 (\pi )/(2), (3\pi )/(2) 0, \pi (\pi )/( )/(4) 0, (\pi )/(2), \pi , (3\pi )/(2)

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Answer from: howell62

0, π, 2π

Nevertheless, that is not an option. I see two possibilities: 1) the options are misswirtten, 2) the domain is not well defined.

If the domain were 0 ≤ θ < 2π, then 2π were excluded of the domain ant the answer would be 0, π.

Explanation:

1) The first solution, θ = 0 is trivial:

sin (0) - tan (0) = 0

0 - 0 = 0

2) For other solutions, work the expression:

sin(θ) + tan (-θ) = 0 ← given

sin (θ) - tan(θ) = 0 ← tan (-θ) = tan(θ)

sin(θ) - sin (θ) / cos(θ) = 0 ← tan(θ) = sin(θ) / cos(θ)

sin (θ) [1 - 1/cos(θ)] = 0 ← common factor sin(θ)

⇒ Any of the two factors can be 0

⇒ sin (θ) = 0 or (1 - 1 / cos(θ) = 0,

sin(θ) = 0 ⇒ θ = 0, π, 2π

1 - 1/cos(θ) = 0 ⇒ 1/cos(θ) = 1 ⇒ cos(θ) = 1 ⇒ θ = 0, 2π

⇒ Solutions are 0, π, and 2π

In fact if you test with any of those values the equation is checked. The only way to exclude one of those solutions is changing the domain.

Answer from: Quest

starting with:

5 + x - 12 = 2x - 7

5 + x - 12 - x = 2x - 7 - x (subtract x from both sides)

5 - 12 = x - 7 (combine and cancel x's from both sides)

5 - 12 + 7 = x - 7 + 7 (add 7 to both sides)

0 = x (add numbers together on both sides)

so in order for the equation to be true, x must be 0.

trying x = -0.5

5 + x - 12 = 2x - 7 (original equation)

5 + -0.5 - 12 = 2(-0.5) - 7 (plug in value of x)

5 + -0.5 - 12 = -1 - 7 (did multiplication)

-7.5 = -8 (did addition on both sides)

since -7.5 is obviously not equal to -8, the value of -0.5 for x is wrong.

trying x = 0

5 + x - 12 = 2x - 7 (original equation)

5 + 0 - 12 = 2(0) - 7 (plug in value of x)

5 + 0 - 12 = 0 - 7 (did multiplication)

-7 = -7 (did addition on both sides)

since -7 equals -7, that demonstrates that the value of 0 for x is correct.

trying x = 1

5 + x - 12 = 2x - 7 (original equation)

5 + 1 - 12 = 2(1) - 7 (plug in value of x)

5 + 1 - 12 = 2 - 7 (did multiplication)

-6 = -5 (did addition on both sides)

since -6 is obviously not equal to -5, the value of 1 for x is wrong.

Answer from: Quest

1. perpendicular

2. parallel

3. perpendicular

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