Given: \[$\sin A = \frac{12}{13}\$\], \[$\hat{A}\$\] Is Obtuse, And \[$\tan B = \frac{4}{3}\$\], \[$\hat{B} \ \textgreater \ 90^{\circ}\$\].Determine These Ratios Without Calculating Values For The Angles:1.

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject in mathematics and has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will explore how to determine trigonometric ratios without calculating the values of the angles.

Given Information

We are given the following information:

• $\sin A = \frac{12}{13}$
• $\hat{A}$ is obtuse
• $\tan B = \frac{4}{3}$
• $\hat{B} > 90^{\circ}$

Understanding the Given Information

From the given information, we can deduce the following:

• Since $\sin A = \frac{12}{13}$, we know that $A$ is an acute angle because the sine of an obtuse angle is always greater than 1.
• Since $\hat{B} > 90^{\circ}$, we know that $B$ is an obtuse angle.
• Since $\tan B = \frac{4}{3}$, we can use the definition of tangent to write $\tan B = \frac{\sin B}{\cos B} = \frac{4}{3}$.

Determining Trigonometric Ratios

We are asked to determine the following ratios without calculating the values of the angles:

1. $\cos A$
2. $\sin B$
3. $\cos B$

Determining $\cos A$

We can use the Pythagorean identity to determine $\cos A$:

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

Since $\sin A = \frac{12}{13}$, we can substitute this value into the equation:

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

Simplifying the equation, we get:

$\frac{144}{169} + \cos^2 A = 1$

Subtracting $\frac{144}{169}$ from both sides, we get:

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

Since $A$ is an acute angle, we know that $\cos A$ is positive. Therefore, we can take the square root of both sides to get:

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

Determining $\sin B$

We can use the definition of tangent to determine $\sin B$:

$\tan B = \frac{\sin B}{\cos B} = \frac{4}{3}$

Since $\tan B = \frac{4}{3}$, we can write:

$\frac{\sin B}{\cos B} = \frac{4}{3}$

We can also use the Pythagorean identity to write:

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

Since $\tan B = \frac{4}{3}$, we can substitute this value into the equation:

$\left(\frac{4}{3}\right)^2 + \cos^2 B = 1$

Simplifying the equation, we get:

$\frac{16}{9} + \cos^2 B = 1$

Subtracting $\frac{16}{9}$ from both sides, we get:

$\cos^2 B = \frac{5}{9}$

Since $B$ is an obtuse angle, we know that $\cos B$ is negative. Therefore, we can take the square root of both sides to get:

$\cos B = -\frac{\sqrt{5}}{3}$

Substituting this value into the equation $\frac{\sin B}{\cos B} = \frac{4}{3}$, we get:

$\sin B = \frac{4}{3} \cdot \left(-\frac{\sqrt{5}}{3}\right)$

Simplifying the equation, we get:

$\sin B = -\frac{4\sqrt{5}}{9}$

Determining $\cos B$

We can use the Pythagorean identity to determine $\cos B$:

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

Since $\sin B = -\frac{4\sqrt{5}}{9}$, we can substitute this value into the equation:

$\left(-\frac{4\sqrt{5}}{9}\right)^2 + \cos^2 B = 1$

Simplifying the equation, we get:

$\frac{80}{81} + \cos^2 B = 1$

Subtracting $\frac{80}{81}$ from both sides, we get:

$\cos^2 B = \frac{1}{81}$

Since $B$ is an obtuse angle, we know that $\cos B$ is negative. Therefore, we can take the square root of both sides to get:

$\cos B = -\frac{1}{9}$

Conclusion

In this article, we have shown how to determine trigonometric ratios without calculating the values of the angles. We have used the given information to determine the values of $\cos A$, $\sin B$, and $\cos B$. We have also used the Pythagorean identity to simplify the equations and find the values of the trigonometric ratios.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Trigonometry for Dummies" by Mary Jane Sterling

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Trigonometry

Final Thoughts

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject in mathematics and has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will answer some frequently asked questions about trigonometry.

Q1: What is the difference between sine, cosine, and tangent?

A1: Sine, cosine, and tangent are three fundamental trigonometric ratios that are used to describe the relationships between the sides and angles of triangles. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, and the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

Q2: How do I remember the order of the trigonometric ratios?

A2: One way to remember the order of the trigonometric ratios is to use the mnemonic "SOH-CAH-TOA". This stands for:

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

Q3: What is the Pythagorean identity?

A3: The Pythagorean identity is a fundamental equation in trigonometry that states:

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

This equation can be used to find the values of the trigonometric ratios.

Q4: How do I use the Pythagorean identity to find the values of the trigonometric ratios?

A4: To use the Pythagorean identity to find the values of the trigonometric ratios, you can substitute the values of the trigonometric ratios into the equation and solve for the unknown value. For example, if you know the value of the sine of an angle, you can use the Pythagorean identity to find the value of the cosine of the angle.

Q5: What is the difference between an acute angle and an obtuse angle?

A5: An acute angle is an angle that is less than 90 degrees, while an obtuse angle is an angle that is greater than 90 degrees.

Q6: How do I determine the values of the trigonometric ratios for an obtuse angle?

A6: To determine the values of the trigonometric ratios for an obtuse angle, you can use the fact that the sine of an obtuse angle is equal to the negative of the sine of its complementary angle, and the cosine of an obtuse angle is equal to the negative of the cosine of its complementary angle.

Q7: What is the difference between the sine and cosine of an angle?

A7: The sine and cosine of an angle are two different trigonometric ratios that describe the relationships between the sides and angles of triangles. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse, while the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

Q8: How do I use the trigonometric ratios to solve problems?

A8: To use the trigonometric ratios to solve problems, you can substitute the values of the trigonometric ratios into the equations and solve for the unknown value. For example, if you know the value of the sine of an angle, you can use the Pythagorean identity to find the value of the cosine of the angle.

Q9: What are some common applications of trigonometry?

A9: Trigonometry has numerous applications in various fields, including physics, engineering, and navigation. Some common applications of trigonometry include:

• Calculating the heights of buildings and mountains
• Determining the distances between objects
• Calculating the angles of triangles
• Solving problems in physics and engineering

Q10: How can I learn more about trigonometry?

A10: There are many resources available to learn more about trigonometry, including textbooks, online tutorials, and practice problems. Some popular resources include:

• Khan Academy: Trigonometry
• MIT OpenCourseWare: Trigonometry
• Trigonometry for Dummies by Mary Jane Sterling

Conclusion

Trigonometry is a fundamental subject in mathematics that has numerous applications in various fields. By understanding the relationships between the sides and angles of triangles, we can solve complex problems and make predictions about the world around us. In this article, we have answered some frequently asked questions about trigonometry and provided resources for further learning. We hope that this article has been helpful in understanding the concepts of trigonometry and has inspired you to learn more about this fascinating subject.