MHB Determining all values of Θ (gr. 11 math)

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To determine all values of Θ for the equation cosΘ = -0.8722, the solutions are 151° and 209°, with the latter derived using the identity cos(360° - Θ) = cos(Θ). For cotΘ = 8.1516, the correct solutions are 7° and 187°, as tan(360° - Θ) does not yield an additional valid angle in this context. The discussion highlights the importance of understanding trigonometric identities to find all possible angle solutions. The confusion arises from misapplying the tangent identity, which does not support the inclusion of 353° as a valid answer.
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8) Use each trigonometric ratio to determine all values of Θ, to the nearest degree if 0°≤ Θ ≤ 360°.

c) cosΘ = -0.8722

So, this is what I did for this question:
Θ=cos^-1(-0.8722)
Θ = 151°

This is correct, according to the textbook. However, it also gives another answer: 209°. How do you get this using the equations? (like sin (360° - Θ) = -sinΘ, tan (180° + Θ) = tanΘ, etc.)

d) cotΘ = 8.1516.

So, here's what I did for this part:
Θ = 7°
tan(180+7)= 187°
tan(360-7) = 353°

HOWEVER, the textbook only has the answers 7° and 187°. Why wouldn't 353° be correct?

Thanks!
 
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For part c) consider the identity:

$$\cos\left(360^{\circ}-\theta\right)=\cos(\theta)$$

For part d), you are essentially trying to assert that:

$$\tan(-\theta)=\tan(\theta)$$

Is this true?
 
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