How do you calculate sec 10 degrees?
The Correct Answer and Explanation is:
To calculate the secant of 10 degrees, we use the following trigonometric identity:
[
\sec(\theta) = \frac{1}{\cos(\theta)}
]
Where (\theta) is the angle, and in this case, (\theta = 10^\circ). The secant function is the reciprocal of the cosine function. Therefore, to find (\sec(10^\circ)), we first need to determine the value of (\cos(10^\circ)) and then take the reciprocal of that value.
Step-by-step Calculation:
- Find (\cos(10^\circ)):
Using a scientific calculator or a trigonometric table, we can find that: [
\cos(10^\circ) \approx 0.9848
] - Take the reciprocal of (\cos(10^\circ)):
Once we have (\cos(10^\circ)), we calculate the secant by taking its reciprocal: [
\sec(10^\circ) = \frac{1}{\cos(10^\circ)} = \frac{1}{0.9848} \approx 1.0154
]
Thus, the secant of 10 degrees is approximately 1.0154.
Explanation:
The secant function is one of the basic trigonometric functions and is closely related to the cosine function. While cosine measures the ratio of the adjacent side to the hypotenuse in a right triangle, secant measures the reciprocal of this ratio. This means that (\sec(\theta)) increases as the angle (\theta) approaches 90 degrees, since (\cos(\theta)) decreases towards zero and its reciprocal grows larger.
The secant function is useful in various applications such as physics, engineering, and calculus, especially when dealing with angles in relation to the unit circle or when performing certain integrations.
In summary, to calculate (\sec(10^\circ)), you simply find the cosine of 10 degrees and take its reciprocal, yielding a value of approximately 1.0154.