Finding the Angles in the Second and Third Quadrants

Oct 13, 2025 by miodeya.com 57 views

Introduction

In this exercise, we will be solving for the angle $\theta$ in three different trigonometric equations without the use of a calculator. We will use the properties of trigonometric functions and the unit circle to find the values of $\theta$ .

Part a) $\cos Q = -\frac{1}{2}, \quad Q \in (0^{\circ}, 360^{\circ}$

To solve for $Q$ , we need to find the angle whose cosine is $-\frac{1}{2}$ . We know that the cosine function is negative in the second and third quadrants. Therefore, we can use the unit circle to find the angles in these quadrants.

The cosine function is negative in the second and third quadrants, so we can use the unit circle to find the angles in these quadrants.

Finding the Angles in the Second and Third Quadrants

In the second quadrant, the cosine function is negative, and the sine function is positive. We can use the unit circle to find the angle whose cosine is $-\frac{1}{2}$ and whose sine is positive.

From the unit circle, we can see that the angle whose cosine is $-\frac{1}{2}$ and whose sine is positive is $Q = 120^{\circ}$ .

In the third quadrant, the cosine function is negative, and the sine function is negative. We can use the unit circle to find the angle whose cosine is $-\frac{1}{2}$ and whose sine is negative.

From the unit circle, we can see that the angle whose cosine is $-\frac{1}{2}$ and whose sine is negative is $Q = 240^{\circ}$ .

Conclusion

Therefore, the solutions to the equation $\cos Q = -\frac{1}{2}$ are $Q = 120^{\circ}$ and $Q = 240^{\circ}$ .

Part b) $2 \sin \theta = \sqrt{3}, \quad \theta \in (0^{\circ}, 360^{\circ}$

To solve for $\theta$ , we need to find the angle whose sine is $\frac{\sqrt{3}}{2}$ . We know that the sine function is positive in the first and second quadrants. Therefore, we can use the unit circle to find the angles in these quadrants.

Finding the Angles in the First and Second Quadrants

In the first quadrant, the sine function is positive, and the cosine function is positive. We can use the unit circle to find the angle whose sine is $\frac{\sqrt{3}}{2}$ and whose cosine is positive.

From the unit circle, we can see that the angle whose sine is $\frac{\sqrt{3}}{2}$ and whose cosine is positive is $\theta = 60^{\circ}$ .

In the second quadrant, the sine function is positive, and the cosine function is negative. We can use the unit circle to find the angle whose sine is $\frac{\sqrt{3}}{2}$ and whose cosine is negative.

From the unit circle, we can see that the angle whose sine is $\frac{\sqrt{3}}{2}$ and whose cosine is negative is $\theta = 120^{\circ}$ .

Conclusion

Therefore, the solutions to the equation $2 \sin \theta = \sqrt{3}$ are $\theta = 60^{\circ}$ and $\theta = 120^{\circ}$ .

Part c)

Unfortunately, there is no part c) in the original problem. However, we can provide some additional information on how to solve trigonometric equations without a calculator.

Solving Trigonometric Equations without a Calculator

To solve trigonometric equations without a calculator, we can use the properties of trigonometric functions and the unit circle. We can also use the inverse trigonometric functions to find the values of the angles.

Using the Inverse Trigonometric Functions

The inverse trigonometric functions are used to find the values of the angles. The inverse sine function, denoted by $\sin^{-1}$ , is used to find the angle whose sine is a given value. The inverse cosine function, denoted by $\cos^{-1}$ , is used to find the angle whose cosine is a given value.

Conclusion

In this exercise, we have solved three trigonometric equations without the use of a calculator. We have used the properties of trigonometric functions and the unit circle to find the values of the angles. We have also used the inverse trigonometric functions to find the values of the angles.

Final Answer

Introduction

In our previous article, we solved three trigonometric equations without the use of a calculator. We used the properties of trigonometric functions and the unit circle to find the values of the angles. In this article, we will answer some frequently asked questions about solving trigonometric equations without a calculator.

Q: What are the most common trigonometric equations that can be solved without a calculator?

A: The most common trigonometric equations that can be solved without a calculator are those that involve the sine, cosine, and tangent functions. These equations can be solved using the properties of trigonometric functions and the unit circle.

Q: How do I use the unit circle to solve trigonometric equations?

A: The unit circle is a circle with a radius of 1 that is centered at the origin of the coordinate plane. The unit circle is used to find the values of the sine, cosine, and tangent functions for angles between 0 and 360 degrees. To use the unit circle to solve trigonometric equations, you need to find the angle whose sine, cosine, or tangent is a given value.

Q: What are the inverse trigonometric functions, and how are they used to solve trigonometric equations?

A: The inverse trigonometric functions are used to find the values of the angles. The inverse sine function, denoted by $\sin^{-1}$ , is used to find the angle whose sine is a given value. The inverse cosine function, denoted by $\cos^{-1}$ , is used to find the angle whose cosine is a given value. The inverse tangent function, denoted by $\tan^{-1}$ , is used to find the angle whose tangent is a given value.

Q: How do I use the inverse trigonometric functions to solve trigonometric equations?

A: To use the inverse trigonometric functions to solve trigonometric equations, you need to find the angle whose sine, cosine, or tangent is a given value. For example, if you have the equation $\sin \theta = \frac{\sqrt{3}}{2}$ , you can use the inverse sine function to find the value of $\theta$ .

Q: What are some common mistakes to avoid when solving trigonometric equations without a calculator?

A: Some common mistakes to avoid when solving trigonometric equations without a calculator include:

Not using the unit circle to find the values of the sine, cosine, and tangent functions.
Not using the inverse trigonometric functions to find the values of the angles.
Not checking the solutions to see if they are in the correct quadrant.
Not using the properties of trigonometric functions to simplify the equations.

Q: How can I practice solving trigonometric equations without a calculator?

A: You can practice solving trigonometric equations without a calculator by working through problems in a textbook or online resource. You can also try solving trigonometric equations on your own using the unit circle and the inverse trigonometric functions.

Conclusion

In this article, we have answered some frequently asked questions about solving trigonometric equations without a calculator. We have discussed the use of the unit circle and the inverse trigonometric functions to solve trigonometric equations. We have also provided some tips for practicing solving trigonometric equations without a calculator.

Final Answer

The final answer to the question of how to solve trigonometric equations without a calculator is to use the unit circle and the inverse trigonometric functions.