Let $\tan A = \frac{5}{12}$ With $A$ In The First Quadrant (Q I). Find $\sec (2A)=$.

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Introduction

In trigonometry, the tangent function is defined as the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle in a right-angled triangle. Given that $\tan A = \frac{5}{12}$, we are asked to find the value of $\sec (2A)$. To solve this problem, we will use the double-angle formula for the secant function and the given value of $\tan A$.

The Double-Angle Formula for Secant

The double-angle formula for the secant function is given by:

$$\sec (2A) = \frac{1}{\cos (2A)}$$

We can use the double-angle formula for cosine to express $\cos (2A)$ in terms of $\cos A$ and $\sin A$:

$$\cos (2A) = 2\cos^2 A - 1$$

Finding $\cos A$ and $\sin A$

Since $\tan A = \frac{5}{12}$, we can use the Pythagorean identity to find the values of $\cos A$ and $\sin A$:

$$\cos^2 A + \sin^2 A = 1$$

We can rewrite this equation as:

$$\cos^2 A = 1 - \sin^2 A$$

Since $\tan A = \frac{5}{12}$, we can write:

$$\tan A = \frac{\sin A}{\cos A} = \frac{5}{12}$$

Cross-multiplying, we get:

$$12\sin A = 5\cos A$$

Squaring both sides, we get:

$$144\sin^2 A = 25\cos^2 A$$

Substituting the expression for $\cos^2 A$ from above, we get:

$$144\sin^2 A = 25(1 - \sin^2 A)$$

Expanding and simplifying, we get:

$$169\sin^2 A = 25$$

Dividing both sides by 169, we get:

$$\sin^2 A = \frac{25}{169}$$

Taking the square root of both sides, we get:

$$\sin A = \frac{5}{13}$$

Now that we have the value of $\sin A$, we can find the value of $\cos A$:

$$\cos A = \frac{12}{13}$$

Finding $\cos (2A)$

Now that we have the values of $\cos A$ and $\sin A$, we can find the value of $\cos (2A)$:

$$\cos (2A) = 2\cos^2 A - 1$$

Substituting the value of $\cos A$, we get:

$$\cos (2A) = 2\left(\frac{12}{13}\right)^2 - 1$$

Expanding and simplifying, we get:

$$\cos (2A) = \frac{288}{169} - 1$$

$$\cos (2A) = \frac{119}{169}$$

Finding $\sec (2A)$

Now that we have the value of $\cos (2A)$, we can find the value of $\sec (2A)$:

$$\sec (2A) = \frac{1}{\cos (2A)}$$

Substituting the value of $\cos (2A)$, we get:

$$\sec (2A) = \frac{1}{\frac{119}{169}}$$

Simplifying, we get:

$$\sec (2A) = \frac{169}{119}$$

Conclusion

In this article, we used the double-angle formula for the secant function and the given value of $\tan A$ to find the value of $\sec (2A)$. We first found the values of $\cos A$ and $\sin A$ using the Pythagorean identity and the given value of $\tan A$. We then used these values to find the value of $\cos (2A)$ and finally the value of $\sec (2A)$. The final answer is $\frac{169}{119}$.

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Q: What is the given value of $\tan A$?

A: The given value of $\tan A$ is $\frac{5}{12}$.

Q: What is the first quadrant (Q I) in the context of this problem?

A: In the context of this problem, the first quadrant (Q I) refers to the region of the unit circle where both the x-coordinate and the y-coordinate are positive.

Q: How do we find the value of $\cos A$ and $\sin A$?

A: We find the values of $\cos A$ and $\sin A$ using the Pythagorean identity and the given value of $\tan A$. Specifically, we use the equation $\tan A = \frac{\sin A}{\cos A} = \frac{5}{12}$ to find the values of $\sin A$ and $\cos A$.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is the equation $\cos^2 A + \sin^2 A = 1$, which relates the values of $\cos A$ and $\sin A$.

Q: How do we use the Pythagorean identity to find the values of $\cos A$ and $\sin A$?

A: We use the Pythagorean identity to find the values of $\cos A$ and $\sin A$ by substituting the expression for $\tan A$ into the equation and solving for $\sin A$ and $\cos A$.

Q: What is the double-angle formula for the secant function?

A: The double-angle formula for the secant function is $\sec (2A) = \frac{1}{\cos (2A)}$.

Q: How do we find the value of $\cos (2A)$?

A: We find the value of $\cos (2A)$ by using the double-angle formula for cosine: $\cos (2A) = 2\cos^2 A - 1$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\frac{169}{119}$.

Q: What is the significance of the first quadrant (Q I) in this problem?

A: The first quadrant (Q I) is significant in this problem because it ensures that the values of $\cos A$ and $\sin A$ are positive, which is necessary for the calculations to be valid.

Q: How do we ensure that the values of $\cos A$ and $\sin A$ are positive?

A: We ensure that the values of $\cos A$ and $\sin A$ are positive by using the fact that $A$ is in the first quadrant (Q I), which means that both the x-coordinate and the y-coordinate are positive.

Q: What is the relationship between the tangent function and the secant function?

A: The tangent function and the secant function are related by the equation $\tan A = \frac{\sin A}{\cos A}$ and $\sec A = \frac{1}{\cos A}$.

Q: How do we use the relationship between the tangent function and the secant function to find the value of $\sec (2A)$?

A: We use the relationship between the tangent function and the secant function to find the value of $\sec (2A)$ by first finding the value of $\cos (2A)$ and then using the double-angle formula for the secant function.

Q: What is the final step in finding the value of $\sec (2A)$?

A: The final step in finding the value of $\sec (2A)$ is to simplify the expression and obtain the final answer.