To find the value of
tan(8), we start with the given information:
- sec(8) = \( \frac{\sqrt{37}}{6} \)
- sin(8) = 0
Firstly, we recall the relationship between the
secant,
cosine, and
sine functions:
- sec(θ) = \( \frac{1}{cos(θ)} \)
- Therefore, if we know that sec(8) = \( \frac{\sqrt{37}}{6} \), we can express this in terms of the cosine function:
cos(8) = \( \frac{6}{\sqrt{37}} \)
Next, we use the Pythagorean identity:
- sin²(θ) + cos²(θ) = 1.
- Substituting in the value of cos(8):
sin²(8) + \( \left( \frac{6}{\sqrt{37}} \right)^2 = 1 \)
Upon squaring cos(8), we find:
sin²(8) + \( \frac{36}{37} = 1 \)
To solve for sin²(8), we subtract:
sin²(8) = 1 – \( \frac{36}{37} = \frac{1}{37} \)
Given that sin(8) = 0, we need to review the situation:
- If sin(8) = 0, this indicates that angle 8 could correspond to 0 degrees or any multiple of 180 degrees, meaning for specific angles, tan(8) would also be undefined or equal to 0.
However, if we just look at the value based on tan(θ) = \( \frac{sin(θ)}{cos(θ)} \), we have:
tan(8) = \( \frac{0}{\frac{6}{\sqrt{37}}} = 0 \)
Therefore, the final answer is:
tan(8) = 0.