If $\theta = \frac{\pi}{4}$, Then Find Exact Values For The Following:- $\sec(\theta$\] Equals $\square$- $\csc(\theta$\] Equals $\square$- $\tan(\theta$\] Equals $\square$-

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Trigonometric Functions: Finding Exact Values for Secant, Cosecant, and Tangent

In trigonometry, there are six fundamental functions: sine, cosine, tangent, secant, cosecant, and cotangent. These functions are used to describe the relationships between the angles and side lengths of triangles. In this article, we will focus on finding exact values for the secant, cosecant, and tangent functions when the angle θ\theta is equal to π4\frac{\pi}{4}.

Recall of Trigonometric Functions

Before we dive into finding the exact values for the secant, cosecant, and tangent functions, let's recall the definitions of these functions.

  • Sine: sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
  • Cosine: cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
  • Tangent: tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
  • Secant: sec⁡(θ)=hypotenuseadjacent\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}}
  • Cosecant: csc⁡(θ)=hypotenuseopposite\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}}

Finding Exact Values for Secant, Cosecant, and Tangent

Now that we have recalled the definitions of the trigonometric functions, let's find the exact values for the secant, cosecant, and tangent functions when the angle θ\theta is equal to π4\frac{\pi}{4}.

Secant Function

To find the exact value of the secant function, we need to find the value of the cosine function. Since θ=π4\theta = \frac{\pi}{4}, we can use the unit circle to find the value of the cosine function.

import math

theta = math.pi / 4
cos_theta = math.cos(theta)
sec_theta = 1 / cos_theta
print("The exact value of the secant function is:", sec_theta)

Using the unit circle, we can see that the cosine function is equal to 12\frac{1}{\sqrt{2}}. Therefore, the secant function is equal to 112=2\frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2}.

Cosecant Function

To find the exact value of the cosecant function, we need to find the value of the sine function. Since θ=π4\theta = \frac{\pi}{4}, we can use the unit circle to find the value of the sine function.

import math

theta = math.pi / 4
sin_theta = math.sin(theta)
csc_theta = 1 / sin_theta
print("The exact value of the cosecant function is:", csc_theta)

Using the unit circle, we can see that the sine function is equal to 12\frac{1}{\sqrt{2}}. Therefore, the cosecant function is equal to 112=2\frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2}.

Tangent Function

To find the exact value of the tangent function, we need to find the value of the sine function and the cosine function. Since θ=π4\theta = \frac{\pi}{4}, we can use the unit circle to find the values of the sine and cosine functions.

import math

theta = math.pi / 4
sin_theta = math.sin(theta)
cos_theta = math.cos(theta)
tan_theta = sin_theta / cos_theta
print("The exact value of the tangent function is:", tan_theta)

Using the unit circle, we can see that the sine function is equal to 12\frac{1}{\sqrt{2}} and the cosine function is equal to 12\frac{1}{\sqrt{2}}. Therefore, the tangent function is equal to 1212=1\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = 1.

In this article, we have found the exact values for the secant, cosecant, and tangent functions when the angle θ\theta is equal to π4\frac{\pi}{4}. We have used the unit circle to find the values of the sine and cosine functions, and then used these values to find the exact values of the secant, cosecant, and tangent functions.

  • Secant Function: sec⁡(π4)=2\sec(\frac{\pi}{4}) = \boxed{\sqrt{2}}
  • Cosecant Function: csc⁡(π4)=2\csc(\frac{\pi}{4}) = \boxed{\sqrt{2}}
  • Tangent Function: tan⁡(π4)=1\tan(\frac{\pi}{4}) = \boxed{1}
    Trigonometric Functions: Q&A

In our previous article, we discussed finding exact values for the secant, cosecant, and tangent functions when the angle θ\theta is equal to π4\frac{\pi}{4}. In this article, we will answer some frequently asked questions about trigonometric functions.

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are two of the most fundamental trigonometric functions. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse, while the cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Q: How do I remember the order of the trigonometric functions?

A: One way to remember the order of the trigonometric functions is to use the mnemonic "SOH-CAH-TOA". This stands for:

  • Sine: Opposite over Hypotenuse
  • Cosine: Adjacent over Hypotenuse
  • Tangent: Opposite over Adjacent

Q: What is the relationship between the secant and cosine functions?

A: The secant function is the reciprocal of the cosine function. This means that if the cosine function is equal to a certain value, the secant function will be equal to the reciprocal of that value.

Q: What is the relationship between the cosecant and sine functions?

A: The cosecant function is the reciprocal of the sine function. This means that if the sine function is equal to a certain value, the cosecant function will be equal to the reciprocal of that value.

Q: How do I find the exact value of the tangent function?

A: To find the exact value of the tangent function, you need to find the values of the sine and cosine functions. You can use the unit circle to find these values, or you can use the definitions of the sine and cosine functions to find them.

Q: What is the value of the tangent function when the angle is equal to π4\frac{\pi}{4}?

A: The value of the tangent function when the angle is equal to π4\frac{\pi}{4} is equal to 1.

Q: What is the value of the secant function when the angle is equal to π4\frac{\pi}{4}?

A: The value of the secant function when the angle is equal to π4\frac{\pi}{4} is equal to 2\sqrt{2}.

Q: What is the value of the cosecant function when the angle is equal to π4\frac{\pi}{4}?

A: The value of the cosecant function when the angle is equal to π4\frac{\pi}{4} is equal to 2\sqrt{2}.

In this article, we have answered some frequently asked questions about trigonometric functions. We have discussed the differences between the sine and cosine functions, how to remember the order of the trigonometric functions, and the relationships between the secant and cosine functions and the cosecant and sine functions. We have also provided examples of how to find the exact values of the tangent, secant, and cosecant functions when the angle is equal to π4\frac{\pi}{4}.

  • Unit Circle: The unit circle is a circle with a radius of 1 that is centered at the origin of the coordinate plane. It is used to find the values of the sine and cosine functions.
  • Trigonometric Identities: Trigonometric identities are equations that are true for all values of the angle. They can be used to find the values of the sine and cosine functions.
  • Trigonometric Functions: Trigonometric functions are mathematical functions that describe the relationships between the angles and side lengths of triangles. They include the sine, cosine, tangent, secant, cosecant, and cotangent functions.
  • Secant Function: sec⁡(π4)=2\sec(\frac{\pi}{4}) = \boxed{\sqrt{2}}
  • Cosecant Function: csc⁡(π4)=2\csc(\frac{\pi}{4}) = \boxed{\sqrt{2}}
  • Tangent Function: tan⁡(π4)=1\tan(\frac{\pi}{4}) = \boxed{1}