Recalling Trigonometric Definitions

Oct 13, 2025 by magicalceylontours.com 106 views

Introduction

In this revision exercise, we will be calculating trigonometric values without the use of a calculator. We will focus on finding the cosine of a reflex angle, given a point on the Cartesian Plane.

Understanding the Problem

We are given a point ${$ M (2, -1) $}$ on the Cartesian Plane, and we need to find the cosine of the angle ${$ \angle XOM = \theta $}$, where ${$ \theta $}$ is a reflex angle.

Recalling Trigonometric Definitions

Before we proceed, let's recall the definitions of sine, cosine, and tangent in a right-angled triangle.

Sine: ${$ \sin \theta = \frac{opposite}{hypotenuse} $}$
Cosine: ${$ \cos \theta = \frac{adjacent}{hypotenuse} $}$
Tangent: ${$ \tan \theta = \frac{opposite}{adjacent} $}$

Finding the Cosine of a Reflex Angle

To find the cosine of a reflex angle, we need to use the fact that the cosine function is periodic with a period of ${$ 2\pi $}$. This means that the cosine of an angle is equal to the cosine of the angle minus ${$ 2\pi $}$.

In this case, we can write ${$ \cos \theta = \cos (2\pi - \theta) $}$.

Using the Point ${$ M (2, -1) $}$

We are given the point ${$ M (2, -1) $}$ on the Cartesian Plane. We can use this point to find the cosine of the angle ${$ \angle XOM = \theta $}$.

To do this, we need to find the coordinates of the point ${$ X $}$ and ${$ O $}$.

The coordinates of the point ${$ X $}$ are ${$ (2, -1) $}$.
The coordinates of the point ${$ O $}$ are ${$ (0, 0) $}$.

Finding the Distance Between Two Points

To find the cosine of the angle ${$ \angle XOM = \theta $}$, we need to find the distance between the points ${$ X $}$ and ${$ O $}$.

The distance between two points ${$ (x_1, y_1) $}$ and ${$ (x_2, y_2) $}$ is given by the formula:

${$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $}$

In this case, the distance between the points ${$ X $}$ and ${$ O $}$ is:

${$ d = \sqrt{(2 - 0)^2 + (-1 - 0)^2} $}$

${$ d = \sqrt{4 + 1} $}$

${$ d = \sqrt{5} $}$

Finding the Cosine of the Angle

Now that we have the distance between the points ${$ X $}$ and ${$ O $}$, we can find the cosine of the angle ${$ \angle XOM = \theta $}$.

The cosine of an angle is given by the formula:

${$ \cos \theta = \frac{adjacent}{hypotenuse} $}$

In this case, the adjacent side is the x-coordinate of the point ${$ X $}$, which is ${$ 2 $}$.

The hypotenuse is the distance between the points ${$ X $}$ and ${$ O $}$, which is ${$ \sqrt{5} $}$.

Therefore, the cosine of the angle ${$ \angle XOM = \theta $}$ is:

${$ \cos \theta = \frac{2}{\sqrt{5}} $}$

Simplifying the Expression

To simplify the expression, we can rationalize the denominator by multiplying the numerator and denominator by ${$ \sqrt{5} $}$.

${$ \cos \theta = \frac{2}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} $}$

${$ \cos \theta = \frac{2\sqrt{5}}{5} $}$

Conclusion

In this revision exercise, we calculated the cosine of a reflex angle without the use of a calculator. We used the point ${$ M (2, -1) $}$ on the Cartesian Plane to find the cosine of the angle ${$ \angle XOM = \theta $}$.

We found the distance between the points ${$ X $}$ and ${$ O $}$ using the distance formula, and then used the cosine formula to find the cosine of the angle.

The final answer is ${$ \frac{2\sqrt{5}}{5} $}$.

Revision Exercise 1: Calculating Trigonometric Values

Introduction

In this revision exercise, we will be calculating trigonometric values without the use of a calculator. We will focus on finding the cosine of a reflex angle, given a point on the Cartesian Plane.

Understanding the Problem

We are given a point ${$ M (2, -1) $}$ on the Cartesian Plane, and we need to find the cosine of the angle ${$ \angle XOM = \theta $}$, where ${$ \theta $}$ is a reflex angle.

Recalling Trigonometric Definitions

Before we proceed, let's recall the definitions of sine, cosine, and tangent in a right-angled triangle.

Sine: ${$ \sin \theta = \frac{opposite}{hypotenuse} $}$
Cosine: ${$ \cos \theta = \frac{adjacent}{hypotenuse} $}$
Tangent: ${$ \tan \theta = \frac{opposite}{adjacent} $}$

Finding the Cosine of a Reflex Angle

In this case, we can write ${$ \cos \theta = \cos (2\pi - \theta) $}$.

Using the Point ${$ M (2, -1) $}$

We are given the point ${$ M (2, -1) $}$ on the Cartesian Plane. We can use this point to find the cosine of the angle ${$ \angle XOM = \theta $}$.

To do this, we need to find the coordinates of the point ${$ X $}$ and ${$ O $}$.

The coordinates of the point ${$ X $}$ are ${$ (2, -1) $}$.
The coordinates of the point ${$ O $}$ are ${$ (0, 0) $}$.

Finding the Distance Between Two Points

To find the cosine of the angle ${$ \angle XOM = \theta $}$, we need to find the distance between the points ${$ X $}$ and ${$ O $}$.

The distance between two points ${$ (x_1, y_1) $}$ and ${$ (x_2, y_2) $}$ is given by the formula:

${$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $}$

In this case, the distance between the points ${$ X $}$ and ${$ O $}$ is:

${$ d = \sqrt{(2 - 0)^2 + (-1 - 0)^2} $}$

${$ d = \sqrt{4 + 1} $}$

${$ d = \sqrt{5} $}$

Finding the Cosine of the Angle

Now that we have the distance between the points ${$ X $}$ and ${$ O $}$, we can find the cosine of the angle ${$ \angle XOM = \theta $}$.

The cosine of an angle is given by the formula:

${$ \cos \theta = \frac{adjacent}{hypotenuse} $}$

In this case, the adjacent side is the x-coordinate of the point ${$ X $}$, which is ${$ 2 $}$.

The hypotenuse is the distance between the points ${$ X $}$ and ${$ O $}$, which is ${$ \sqrt{5} $}$.

Therefore, the cosine of the angle ${$ \angle XOM = \theta $}$ is:

${$ \cos \theta = \frac{2}{\sqrt{5}} $}$

Simplifying the Expression

To simplify the expression, we can rationalize the denominator by multiplying the numerator and denominator by ${$ \sqrt{5} $}$.

${$ \cos \theta = \frac{2}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} $}$

${$ \cos \theta = \frac{2\sqrt{5}}{5} $}$

Conclusion

Introduction

In this Q&A article, we will be answering some common questions related to the revision exercise on calculating trigonometric values without the use of a calculator.

Q: What is the formula for finding the cosine of an angle?

A: The formula for finding the cosine of an angle is:

[$ \cos \theta = \frac{adjacent}{hypotenuse} $}$

Q: How do I find the adjacent side and the hypotenuse in a right-angled triangle?

A: To find the adjacent side and the hypotenuse in a right-angled triangle, you need to use the Pythagorean theorem, which states that:

${$ a^2 + b^2 = c^2 $}$

where ${$ a $}$ and ${$ b $}$ are the lengths of the two sides that form the right angle, and ${$ c $}$ is the length of the hypotenuse.

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

A: The sine, cosine, and tangent functions are all trigonometric functions that are used to describe the relationships between the angles and sides of a right-angled triangle.

Sine: ${$ \sin \theta = \frac{opposite}{hypotenuse} $}$
Cosine: ${$ \cos \theta = \frac{adjacent}{hypotenuse} $}$
Tangent: ${$ \tan \theta = \frac{opposite}{adjacent} $}$

Q: How do I find the cosine of a reflex angle?

A: To find the cosine of a reflex angle, you need to use the fact that the cosine function is periodic with a period of ${$ 2\pi $}$. This means that the cosine of an angle is equal to the cosine of the angle minus ${$ 2\pi $}$.

In this case, we can write ${$ \cos \theta = \cos (2\pi - \theta) $}$.

Q: What is the formula for finding the distance between two points on the Cartesian Plane?

A: The formula for finding the distance between two points ${$ (x_1, y_1) $}$ and ${$ (x_2, y_2) $}$ on the Cartesian Plane is:

${$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $}$

Q: How do I use the distance formula to find the cosine of an angle?

A: To use the distance formula to find the cosine of an angle, you need to find the distance between the points ${$ X $}$ and ${$ O $}$ on the Cartesian Plane, and then use the cosine formula to find the cosine of the angle.

The cosine of an angle is given by the formula:

${$ \cos \theta = \frac{adjacent}{hypotenuse} $}$

In this case, the adjacent side is the x-coordinate of the point ${$ X $}$, and the hypotenuse is the distance between the points ${$ X $}$ and ${$ O $}$.

Q: What is the final answer to the revision exercise?

A: The final answer to the revision exercise is ${$ \frac{2\sqrt{5}}{5} $}$.

Conclusion

In this Q&A article, we have answered some common questions related to the revision exercise on calculating trigonometric values without the use of a calculator. We have covered topics such as the formula for finding the cosine of an angle, the difference between the sine, cosine, and tangent functions, and how to use the distance formula to find the cosine of an angle.

We hope that this Q&A article has been helpful in clarifying any doubts that you may have had about the revision exercise. If you have any further questions, please do not hesitate to ask.

Revision Exercise 1: Calculating Trigonometric Values - Q&A

Introduction

In this Q&A article, we will be answering some common questions related to the revision exercise on calculating trigonometric values without the use of a calculator.

Q: What is the formula for finding the cosine of an angle?

A: The formula for finding the cosine of an angle is:

${$ \cos \theta = \frac{adjacent}{hypotenuse} $}$

Q: How do I find the adjacent side and the hypotenuse in a right-angled triangle?

A: To find the adjacent side and the hypotenuse in a right-angled triangle, you need to use the Pythagorean theorem, which states that:

${$ a^2 + b^2 = c^2 $}$

where ${$ a $}$ and ${$ b $}$ are the lengths of the two sides that form the right angle, and ${$ c $}$ is the length of the hypotenuse.

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

A: The sine, cosine, and tangent functions are all trigonometric functions that are used to describe the relationships between the angles and sides of a right-angled triangle.

Sine: ${$ \sin \theta = \frac{opposite}{hypotenuse} $}$
Cosine: ${$ \cos \theta = \frac{adjacent}{hypotenuse} $}$
Tangent: ${$ \tan \theta = \frac{opposite}{adjacent} $}$

Q: How do I find the cosine of a reflex angle?

In this case, we can write ${$ \cos \theta = \cos (2\pi - \theta) $}$.

Q: What is the formula for finding the distance between two points on the Cartesian Plane?

A: The formula for finding the distance between two points ${$ (x_1, y_1) $}$ and ${$ (x_2, y_2) $}$ on the Cartesian Plane is:

${$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $}$

Q: How do I use the distance formula to find the cosine of an angle?

The cosine of an angle is given by the formula:

${$ \cos \theta = \frac{adjacent}{hypotenuse} $}$

In this case, the adjacent side is the x-coordinate of the point ${$ X $}$, and the hypotenuse is the distance between the points ${$ X $}$ and ${$ O $}$.

Q: What is the final answer to the revision exercise?

A: The final answer to the revision exercise is ${$ \frac{2\sqrt{5}}{5} $}$.

Conclusion

We hope that this Q&A article has been helpful in clarifying any doubts that you may have had about the revision exercise. If you have any further questions, please do not hesitate to ask.