Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact values of the sine, cosine, and tangent of the angle. $$165^{\circ}=135^{\circ}+30^{\circ}$$

Step-by-Step Solution

Answer

The exact values of sine, cosine and tangent of the given angle are \(sin 165^\circ = (sqrt{6}-sqrt{2})/4\), \(cos 165^\circ = -(sqrt{6}+sqrt{2})/4\) and \(tan 165^\circ = -1\).

Step 1 - Decompose the Angle

Given angle \(165^\circ\) is decomposed as sum of two known angles: \(165^\circ=135^\circ+30^\circ\). Here \(135^\circ\) is a multiple of \(45^\circ\) and \(30^\circ\) is standard angle. Both of these angles have known sine, cosine, and tangent values.

Step 2 - Find Sine

Sine of given angle can be found by using sine addition formula \(sin(\alpha+\beta)=sin\alpha cos\beta+cos\alpha sin\beta\). Substituting \(\alpha = 135^\circ\) and \(\beta = 30^\circ\) into the formula, we get \(sin 165 = sin 135 cos 30 + cos 135 sin 30 = cos 45 sqrt{3}/2 - sin 45/2 = sqrt{2}/2 sqrt{3}/2 - sqrt{2}/4 = (sqrt{6}-sqrt{2})/4.\)

Step 3 - Find Cosine

Cosine of given angle can be found by using cosine addition formula \(cos(\alpha+\beta)=cos\alpha cos\beta - sin \alpha sin \beta\). Substituting \(\alpha = 135^\circ\) and \(\beta = 30^\circ\) into the formula, we get \(cos 165 = cos 135 cos 30 - sin 135 sin 30 = -cos 45 sqrt{3}/2 - sin 45/2 = -sqrt{2}/2 sqrt{3}/2 - sqrt{2}/4 = -(sqrt{6}+sqrt{2})/4\).

Step 4 - Find Tangent

Using the definition, the tangent of an angle is the ratio of its sine to its cosine, we find the tangent \(\tan 165^\circ=\frac{sin 165}{cos 165} = \frac{(sqrt{6}-sqrt{2})/4}{-(sqrt{6}+sqrt{2})/4} = -1.\)