How do you calculate sec 10 degrees?
The Correct Answer and Explanation is:
To calculate ( \sec 10^\circ ), which represents the secant of ( 10^\circ ), you need to understand the concept of secant and how it relates to cosine. In trigonometry, the secant of an angle is the reciprocal of the cosine of that angle. Mathematically, this is expressed as:
[
\sec \theta = \frac{1}{\cos \theta}
]
So, for ( \sec 10^\circ ):
[
\sec 10^\circ = \frac{1}{\cos 10^\circ}
]
To calculate this value, you need to first determine ( \cos 10^\circ ). Using a scientific calculator or a trigonometric table, you can find that:
[
\cos 10^\circ \approx 0.9848
]
Now, substitute this value into the equation:
[
\sec 10^\circ = \frac{1}{0.9848} \approx 1.0154
]
Thus, the value of ( \sec 10^\circ ) is approximately ( 1.0154 ).
Explanation
In trigonometry, secant is one of the six primary trigonometric functions, and it is related to cosine. While cosine represents the ratio of the adjacent side to the hypotenuse in a right triangle, secant represents the ratio of the hypotenuse to the adjacent side. Secant is especially useful when dealing with angles in different quadrants on the unit circle and is often used in calculus, physics, and engineering when dealing with wave functions, oscillations, and periodic phenomena.
The secant function is rarely given directly in calculators, which typically have sine, cosine, and tangent functions, but it can be calculated by taking the reciprocal of cosine. When you calculate ( \sec 10^\circ ), you are essentially asking, “What is the ratio of the hypotenuse to the adjacent side for an angle of ( 10^\circ )?” Using ( \cos 10^\circ ), which is the ratio of the adjacent side to the hypotenuse, and then taking the reciprocal of this value provides the answer.
In summary, by finding ( \cos 10^\circ ) and using the relationship ( \sec 10^\circ = \frac{1}{\cos 10^\circ} ), you determine that ( \sec 10^\circ \approx 1.0154 ). This approach applies to any angle when calculating secant, provided you can find or approximate its cosine.