Trigonometric Ratios and Their Definitions

Oct 13, 2025 by talynt.org 106 views

Understanding the Problem

When dealing with trigonometric ratios, it's essential to identify the type of triangle and the given information to determine the appropriate equation to use. In this scenario, we're given the opposite side (6 meters) and the value of $\tan \left(60^{\circ}\right)$ , which is equal to 1.732. We need to find the equation that would be used to determine the adjacent side.

Trigonometric Ratios and Their Definitions

Before we dive into the problem, let's quickly review the definitions of the three basic trigonometric ratios: sine, cosine, and tangent.

Sine (sin): The ratio of the opposite side to the hypotenuse in a right-angled triangle.
Cosine (cos): The ratio of the adjacent side to the hypotenuse in a right-angled triangle.
Tangent (tan): The ratio of the opposite side to the adjacent side in a right-angled triangle.

Analyzing the Given Information

We're given the value of $\tan \left(60^{\circ}\right)$ , which is equal to 1.732. This means that the ratio of the opposite side to the adjacent side is 1.732. We also know that the opposite side is 6 meters.

Finding the Equation

To find the equation that would be used to determine the adjacent side, we need to use the definition of tangent. Since we know the value of $\tan \left(60^{\circ}\right)$ and the opposite side, we can use the following equation:

\tan \left(60^{\circ}\right)=\frac{\text{opposite}}{\text{adjacent}}

Substituting the given values, we get:

1.732=\frac{6}{\text{adjacent}}

Solving for the Adjacent Side

To find the adjacent side, we can rearrange the equation to isolate the adjacent side:

\text{adjacent}=\frac{6}{1.732}

Calculating the Adjacent Side

Now, let's calculate the value of the adjacent side:

\text{adjacent}=\frac{6}{1.732}\approx3.464

Conclusion

In conclusion, the equation that would be used to find the adjacent side is:

\tan \left(60^{\circ}\right)=\frac{\text{opposite}}{\text{adjacent}}

Substituting the given values, we get:

1.732=\frac{6}{\text{adjacent}}

Solving for the adjacent side, we get:

\text{adjacent}=\frac{6}{1.732}\approx3.464

Therefore, the equation that would be used to find the adjacent side is option A: $\tan \left(60^{\circ}\right)=\frac{\text{hypotenuse}}{6}$ is incorrect, the correct equation is $\tan \left(60^{\circ}\right)=\frac{\text{opposite}}{\text{adjacent}}$

Understanding the Problem

Trigonometric Ratios and Their Definitions

Before we dive into the problem, let's quickly review the definitions of the three basic trigonometric ratios: sine, cosine, and tangent.

Sine (sin): The ratio of the opposite side to the hypotenuse in a right-angled triangle.
Cosine (cos): The ratio of the adjacent side to the hypotenuse in a right-angled triangle.
Tangent (tan): The ratio of the opposite side to the adjacent side in a right-angled triangle.

Analyzing the Given Information

Finding the Equation

\tan \left(60^{\circ}\right)=\frac{\text{opposite}}{\text{adjacent}}

Substituting the given values, we get:

1.732=\frac{6}{\text{adjacent}}

Solving for the Adjacent Side

To find the adjacent side, we can rearrange the equation to isolate the adjacent side:

\text{adjacent}=\frac{6}{1.732}

Calculating the Adjacent Side

Now, let's calculate the value of the adjacent side:

\text{adjacent}=\frac{6}{1.732}\approx3.464

Conclusion

In conclusion, the equation that would be used to find the adjacent side is:

\tan \left(60^{\circ}\right)=\frac{\text{opposite}}{\text{adjacent}}

Substituting the given values, we get:

1.732=\frac{6}{\text{adjacent}}

Solving for the adjacent side, we get:

\text{adjacent}=\frac{6}{1.732}\approx3.464

Q&A

Q: What is the definition of tangent in a right-angled triangle?

A: The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side.

Q: What is the given value of $\tan \left(60^{\circ}\right)$ ?

A: The given value of $\tan \left(60^{\circ}\right)$ is 1.732.

Q: What is the opposite side given in the problem?

A: The opposite side given in the problem is 6 meters.

Q: What is the equation that would be used to find the adjacent side?

A: The equation that would be used to find the adjacent side is:

\tan \left(60^{\circ}\right)=\frac{\text{opposite}}{\text{adjacent}}

Q: What is the value of the adjacent side?

A: The value of the adjacent side is approximately 3.464 meters.

Q: Which option is correct?

A: The correct option is $\tan \left(60^{\circ}\right)=\frac{\text{opposite}}{\text{adjacent}}$

Q: Why is option A incorrect?

A: Option A is incorrect because it states that $\tan \left(60^{\circ}\right)=\frac{\text{hypotenuse}}{6}$ , which is not the correct equation to use in this scenario.

Final Thoughts

In conclusion, the equation that would be used to find the adjacent side is $\tan \left(60^{\circ}\right)=\frac{\text{opposite}}{\text{adjacent}}$ . This equation is based on the definition of tangent in a right-angled triangle. By substituting the given values, we can solve for the adjacent side and find its value.