Use trigonometric identities and compound angle formulas to calculate the exact value of \cot\left(\frac{13\pi}{6}\right). Show at least four lines of work for full marks. [4 Marks]
The Correct Answer and Explanation is:
To calculate the exact value of cot(13π6)\cot\left(\frac{13\pi}{6}\right)cot(613π), begin by expressing the angle in terms of a coterminal angle within [0,2π][0, 2\pi][0,2π].
Step 1: Find a coterminal angle between 000 and 2π2\pi2π13π6−2π=13π6−12π6=π6\frac{13\pi}{6} – 2\pi = \frac{13\pi}{6} – \frac{12\pi}{6} = \frac{\pi}{6}613π−2π=613π−612π=6π
Step 2: Use the coterminal identitycot(13π6)=cot(π6)\cot\left(\frac{13\pi}{6}\right) = \cot\left(\frac{\pi}{6}\right)cot(613π)=cot(6π)
Step 3: Use the definition of cotangent in terms of sine and cosinecot(π6)=cos(π6)sin(π6)\cot\left(\frac{\pi}{6}\right) = \frac{\cos\left(\frac{\pi}{6}\right)}{\sin\left(\frac{\pi}{6}\right)}cot(6π)=sin(6π)cos(6π)
Step 4: Use known exact valuescos(π6)=32,sin(π6)=12\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}, \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}cos(6π)=23,sin(6π)=21cot(π6)=3/21/2=3\cot\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}cot(6π)=1/23/2=3
Final Answer:cot(13π6)=3\cot\left(\frac{13\pi}{6}\right) = \sqrt{3}cot(613π)=3
Explanation
The cotangent of an angle that exceeds 2π2\pi2π can be simplified by identifying a coterminal angle. A coterminal angle shares the same terminal side on the unit circle as the original angle. To find it, subtract 2π2\pi2π from 13π6\frac{13\pi}{6}613π. This operation yields π6\frac{\pi}{6}6π, which lies within the principal interval from 000 to 2π2\pi2π. This implies that the trigonometric value of cot(13π6)\cot\left(\frac{13\pi}{6}\right)cot(613π) is identical to that of cot(π6)\cot\left(\frac{\pi}{6}\right)cot(6π).
The cotangent function, defined as the ratio of cosine to sine, leads to a straightforward computation. The unit circle values for cos(π6)\cos\left(\frac{\pi}{6}\right)cos(6π) and sin(π6)\sin\left(\frac{\pi}{6}\right)sin(6π) are commonly memorized or referenced from standard trigonometric tables. Specifically, the cosine of π6\frac{\pi}{6}6π is 32\frac{\sqrt{3}}{2}23, and the sine is 12\frac{1}{2}21. Dividing these values gives:cos(π6)sin(π6)=3/21/2=3\frac{\cos\left(\frac{\pi}{6}\right)}{\sin\left(\frac{\pi}{6}\right)} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}sin(6π)cos(6π)=1/23/2=3
Thus, the exact value of cot(13π6)\cot\left(\frac{13\pi}{6}\right)cot(613π) is 3\sqrt{3}3. This result showcases the usefulness of understanding coterminal angles and fundamental trigonometric identities, allowing for efficient and accurate evaluations of trigonometric expressions.
