How to Find the Exact Value of Secant 3π/4?
Secant is one of the fundamental trigonometric functions that arises in various mathematical and scientific applications. When dealing with trigonometry, it is essential to understand how to calculate the exact values of these functions. In this article, we will focus on finding the exact value of secant for the angle 3π/4.
Before delving into the specific calculation, it is crucial to define the secant function. Secant is the reciprocal of the cosine function, so we can express it in terms of cosine: sec(θ) = 1/cos(θ).
To find the exact value of secant 3π/4, we need to determine the cosine value of the same angle. The key to solving this is recognizing the related angles and special triangles in the unit circle.
How can we determine the cosine of 3π/4?
To find the cosine of 3π/4, we can use the unit circle. The angle 3π/4 corresponds to the point (-√2/2, -√2/2) on the unit circle. Hence, cos(3π/4) = -√2/2.
Now that we know the cosine value, we can find the exact value of secant 3π/4.
How to calculate the secant of 3π/4?
Since secant is the reciprocal of cosine, we can calculate secant 3π/4 by taking the reciprocal of cos(3π/4). Therefore, sec(3π/4) = 1/(-√2/2).
To simplify this expression, we multiply the numerator and denominator by 2/√2 to rationalize the denominator.
Sec(3π/4) = 1/(-√2/2) * (2/√2) = -2/√2.
However, the value of sec(θ) is typically expressed without radicals in the denominator. To rationalize the denominator fully, we multiply both the numerator and denominator by √2.
Sec(3π/4) = -2/√2 * √2/√2 = -2√2/2.
Finally, we can simplify the expression further by canceling out the common factor of 2.
What is the exact value of secant 3π/4?
The exact value of secant 3π/4 is -√2.
Congratulations! You now know how to find the exact value of secant for the angle 3π/4. However, you might still have some related questions. Let’s address a few of those:
Related FAQs:
1. What is the unit circle?
The unit circle is a circle with a radius of 1 unit centered at the origin (0, 0) in a Cartesian coordinate system.
2. How can we determine the values of trigonometric functions using the unit circle?
By locating an angle on the unit circle and finding the corresponding coordinates, we can determine the values of trigonometric functions for that angle.
3. What are the values of cosine in each quadrant of the unit circle?
In the first quadrant, cosine is positive. In the second quadrant, only sine is positive. In the third quadrant, both sine and cosine are negative. In the fourth quadrant, only cosine is positive.
4. What is the reciprocal of a trigonometric function?
The reciprocal of a trigonometric function is obtained by flipping the ratio or inverting it. For example, the reciprocal of sine is cosecant.
5. How do we rationalize the denominator of a fraction?
To rationalize the denominator of a fraction, we multiply the numerator and denominator by the conjugate of the denominator. This eliminates any radicals from the denominator.
6. What are the related angles on the unit circle?
Related angles are angles that have the same reference angle but differ by an integer multiple of 2π.
7. Can we find the exact value of secant for all angles?
No, the exact value of secant can only be found for certain angles that correspond to neat points on the unit circle or triangles.
8. How is secant related to cosine?
Secant is the reciprocal of cosine. It is calculated by taking the inverse of cosine.
9. What does secant represent in terms of triangles?
Secant represents the ratio of the hypotenuse to the adjacent side of a right-angled triangle.
10. What is the period of the secant function?
The period of the secant function is 2π, which means it repeats its values after every 2π radians.
11. How does the secant function behave for angles close to 0 and π?
The secant function approaches positive infinity as the angle approaches either 0 or π.
12. Are there any real-life applications of secant?
Secant and other trigonometric functions are extensively used in fields like physics, engineering, navigation, and architecture to model and solve real-world problems.