For The Questions Below, { \hat{A}$}$ Is Acute. A) If { \sin A = \frac{4}{5}$}$, Find The Value Of { \cot A + \tan A$}$.b) If { \sin A = \frac{3}{8}$} , F I N D T H E V A L U E O F \[ , Find The Value Of \[ , F In D T H E V A L U Eo F \[ \operatorname{cosec} A - \sec

Trigonometric Identities and Formulas: Solving Acute Angle Problems

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. In this article, we will explore two problems involving acute angles and trigonometric identities. We will use the given information to find the values of various trigonometric functions and expressions.

Problem a: Finding the Value of cot A + tan A

Given Information

We are given that $\sin A = \frac{4}{5}$, where $A$ is an acute angle.

Objective

Our objective is to find the value of $\cot A + \tan A$.

Solution

To solve this problem, we will use the definitions of cotangent and tangent in terms of sine and cosine.

$\cot A = \frac{\cos A}{\sin A}$

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

We can rewrite the expression $\cot A + \tan A$ as:

$\cot A + \tan A = \frac{\cos A}{\sin A} + \frac{\sin A}{\cos A}$

To add these fractions, we need a common denominator, which is $\sin A \cos A$. Therefore, we can rewrite the expression as:

$\cot A + \tan A = \frac{\cos^2 A + \sin^2 A}{\sin A \cos A}$

Using the Pythagorean identity $\cos^2 A + \sin^2 A = 1$, we can simplify the numerator:

$\cot A + \tan A = \frac{1}{\sin A \cos A}$

Now, we can use the given information $\sin A = \frac{4}{5}$ to find the value of $\cos A$.

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

$\cos^2 A = 1 - \left(\frac{4}{5}\right)^2$

$\cos^2 A = 1 - \frac{16}{25}$

$\cos^2 A = \frac{9}{25}$

Taking the square root of both sides, we get:

$\cos A = \pm \frac{3}{5}$

Since $A$ is an acute angle, we know that $\cos A$ is positive. Therefore, we can write:

$\cos A = \frac{3}{5}$

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

$\sin A \cos A = \frac{4}{5} \cdot \frac{3}{5}$

$\sin A \cos A = \frac{12}{25}$

Finally, we can find the value of $\cot A + \tan A$:

$\cot A + \tan A = \frac{1}{\sin A \cos A}$

$\cot A + \tan A = \frac{1}{\frac{12}{25}}$

$\cot A + \tan A = \frac{25}{12}$

Problem b: Finding the Value of cosec A - sec A

Given Information

We are given that $\sin A = \frac{3}{8}$, where $A$ is an acute angle.

Objective

Our objective is to find the value of $\cosec A - \sec A$.

Solution

To solve this problem, we will use the definitions of cosecant and secant in terms of sine and cosine.

$\cosec A = \frac{1}{\sin A}$

$\sec A = \frac{1}{\cos A}$

We can rewrite the expression $\cosec A - \sec A$ as:

$\cosec A - \sec A = \frac{1}{\sin A} - \frac{1}{\cos A}$

To subtract these fractions, we need a common denominator, which is $\sin A \cos A$. Therefore, we can rewrite the expression as:

$\cosec A - \sec A = \frac{\cos A - \sin A}{\sin A \cos A}$

Now, we can use the given information $\sin A = \frac{3}{8}$ to find the value of $\cos A$.

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

$\cos^2 A = 1 - \left(\frac{3}{8}\right)^2$

$\cos^2 A = 1 - \frac{9}{64}$

$\cos^2 A = \frac{55}{64}$

Taking the square root of both sides, we get:

$\cos A = \pm \frac{\sqrt{55}}{8}$

Since $A$ is an acute angle, we know that $\cos A$ is positive. Therefore, we can write:

$\cos A = \frac{\sqrt{55}}{8}$

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

$\sin A \cos A = \frac{3}{8} \cdot \frac{\sqrt{55}}{8}$

$\sin A \cos A = \frac{3\sqrt{55}}{64}$

Finally, we can find the value of $\cosec A - \sec A$:

$\cosec A - \sec A = \frac{\cos A - \sin A}{\sin A \cos A}$

$\cosec A - \sec A = \frac{\frac{\sqrt{55}}{8} - \frac{3}{8}}{\frac{3\sqrt{55}}{64}}$

$\cosec A - \sec A = \frac{\frac{\sqrt{55} - 3}{8}}{\frac{3\sqrt{55}}{64}}$

$\cosec A - \sec A = \frac{8(\sqrt{55} - 3)}{8 \cdot 3\sqrt{55}}$

$\cosec A - \sec A = \frac{\sqrt{55} - 3}{3\sqrt{55}}$

$\cosec A - \sec A = \frac{\sqrt{55} - 3}{3\sqrt{55}} \cdot \frac{\sqrt{55}}{\sqrt{55}}$

$\cosec A - \sec A = \frac{(\sqrt{55})^2 - 3\sqrt{55}}{3(\sqrt{55})^2}$

$\cosec A - \sec A = \frac{55 - 3\sqrt{55}}{3 \cdot 55}$

$\cosec A - \sec A = \frac{55 - 3\sqrt{55}}{165}$

$\cosec A - \sec A = \frac{55}{165} - \frac{3\sqrt{55}}{165}$

$$

In our previous article, we explored two problems involving acute angles and trigonometric identities. We used the given information to find the values of various trigonometric functions and expressions. In this article, we will answer some frequently asked questions related to these problems.

Q: What is the difference between cotangent and tangent?

A: Cotangent and tangent are two trigonometric functions that are defined in terms of sine and cosine. Cotangent is the ratio of the adjacent side to the opposite side in a right triangle, while tangent is the ratio of the opposite side to the adjacent side.

Q: How do you simplify the expression cot A + tan A?

A: To simplify the expression cot A + tan A, we can use the definitions of cotangent and tangent in terms of sine and cosine. We can rewrite the expression as:

cot A + tan A = (cos A / sin A) + (sin A / cos A)

Using the Pythagorean identity cos^2 A + sin^2 A = 1, we can simplify the numerator:

cot A + tan A = (cos^2 A + sin^2 A) / (sin A cos A)

cot A + tan A = 1 / (sin A cos A)

Q: How do you find the value of cosec A - sec A?

A: To find the value of cosec A - sec A, we can use the definitions of cosecant and secant in terms of sine and cosine. We can rewrite the expression as:

cosec A - sec A = (1 / sin A) - (1 / cos A)

Using the given information sin A = 3/8, we can find the value of cos A:

cos^2 A = 1 - sin^2 A

cos^2 A = 1 - (3/8)^2

cos^2 A = 1 - 9/64

cos^2 A = 55/64

Taking the square root of both sides, we get:

cos A = ±√55/8

Since A is an acute angle, we know that cos A is positive. Therefore, we can write:

cos A = √55/8

Now, we can find the value of sin A cos A:

sin A cos A = (3/8) * (√55/8)

sin A cos A = 3√55/64

Finally, we can find the value of cosec A - sec A:

cosec A - sec A = (cos A - sin A) / (sin A cos A)

cosec A - sec A = (√55/8 - 3/8) / (3√55/64)

cosec A - sec A = (√55 - 3) / (3√55)

Q: What is the relationship between trigonometric functions and right triangles?

A: Trigonometric functions are defined in terms of the ratios of the sides of a right triangle. The sine of an angle is the ratio of the opposite side to the hypotenuse, the cosine of an angle is the ratio of the adjacent side to the hypotenuse, and the tangent of an angle is the ratio of the opposite side to the adjacent side.

Q: How do you use trigonometric identities to simplify expressions?

A: Trigonometric identities are equations that relate different trigonometric functions. We can use these identities to simplify expressions by substituting one function for another. For example, we can use the Pythagorean identity cos^2 A + sin^2 A = 1 to simplify the expression cot A + tan A.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• Pythagorean identity: cos^2 A + sin^2 A = 1
• Sum and difference formulas: sin(A + B) = sin A cos B + cos A sin B, cos(A + B) = cos A cos B - sin A sin B
• Double-angle formulas: sin(2A) = 2 sin A cos A, cos(2A) = cos^2 A - sin^2 A

These identities can be used to simplify expressions and solve problems involving trigonometric functions.