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Chapter 5: Problem 26

Find the exact values of the sine, cosine, and tangent of the angle. $$-105^{\circ}$$

Short Answer

Expert verified
The exact values of the sine, cosine, and tangent of the angle \( -105^{\circ} \) are: \(\sin(-105^{\circ}) = -0.5\), \(\cos(-105^{\circ}) = -(\sqrt{6}+\sqrt{2})/4\), and \(\tan(-105^{\circ}) = 2(\sqrt{6}-\sqrt{2})/(\sqrt{6}+\sqrt{2})\) respectively.

Step by step solution

01

Convert to Positive Angle

First convert the negative angle to a positive one by adding \(360^{\circ}\). This gives us \(-105^{\circ} + 360^{\circ} = 255^{\circ}\). The equivalent angle in the first quadrant would be calculated as follows: \(360^{\circ} - 255^{\circ} = 105^{\circ}\). The sine and cosine are negatives of each other. The tangent is the same.
02

Find Sine Value

The sine value of \(105^{\circ}\) can be found using the unit circle or by using the identity for sine of a sum of angles. Here we will use the unit circle. The sine value is negative and is equal to \(-\sin(105^{\circ}) = -\sin(30^{\circ}) = -0.5\).
03

Find Cosine Value

We can now find the cosine value, which is negative in the second quadrant. The cosine value is equal to \(-\cos(75^{\circ}) \). Since \(\cos(75^{\circ})\) is \(\frac{\sqrt{6}+\sqrt{2}}{4}\), the cosine value for \(255^{\circ}\) is \(-\frac{\sqrt{6}+\sqrt{2}}{4}\).
04

Find Tangent Value

We can find the tangent value, which is \(\frac{\text{sin}(255^{\circ})}{\text{cos}(255^{\circ})}\). Therefore, the tangent value is \(\frac{-0.5}{-\frac{\sqrt{6}+\sqrt{2}}{4}}\), which simplifies to \(2\frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}+\sqrt{2}}\).

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