Write Down The Exact Value Of:a) Cos ⁡ 45 ∘ \cos 45^{\circ} Cos 4 5 ∘ B) Sin ⁡ 90 ∘ \sin 90^{\circ} Sin 9 0 ∘ C) Tan ⁡ 60 ∘ \tan 60^{\circ} Tan 6 0 ∘

Introduction

Trigonometry is a branch of mathematics that deals with the study of triangles, particularly right-angled triangles. It involves the use of trigonometric functions such as sine, cosine, and tangent to solve problems related to the lengths and angles of triangles. In this article, we will focus on finding the exact values of $\cos 45^{\circ}$, $\sin 90^{\circ}$, and $\tan 60^{\circ}$.

Recall of Basic Trigonometric Identities

Before we proceed, let's recall some basic trigonometric identities that will be useful in finding the exact values of the given trigonometric functions.

• Pythagorean Identity: $\sin^2 x + \cos^2 x = 1$
• Quotient Identity: $\tan x = \frac{\sin x}{\cos x}$
• Reciprocal Identity: $\csc x = \frac{1}{\sin x}$, $\sec x = \frac{1}{\cos x}$, $\cot x = \frac{1}{\tan x}$

Exact Value of $\cos 45^{\circ}$

To find the exact value of $\cos 45^{\circ}$, we can use the Pythagorean Identity.

$\cos^2 45^{\circ} + \sin^2 45^{\circ} = 1$

Since $\sin 45^{\circ} = \cos 45^{\circ}$, we can substitute $\sin 45^{\circ}$ with $\cos 45^{\circ}$ in the above equation.

$\cos^2 45^{\circ} + \cos^2 45^{\circ} = 1$

Combine like terms:

$2\cos^2 45^{\circ} = 1$

Divide both sides by 2:

$\cos^2 45^{\circ} = \frac{1}{2}$

Take the square root of both sides:

$\cos 45^{\circ} = \pm \sqrt{\frac{1}{2}}$

Since the cosine function is positive in the first quadrant, we take the positive square root:

$\cos 45^{\circ} = \frac{\sqrt{2}}{2}$

Exact Value of $\sin 90^{\circ}$

To find the exact value of $\sin 90^{\circ}$, we can use the Pythagorean Identity.

$\sin^2 90^{\circ} + \cos^2 90^{\circ} = 1$

Since $\sin 90^{\circ} = 1$ and $\cos 90^{\circ} = 0$, we can substitute these values in the above equation.

$1^2 + 0^2 = 1$

Simplify:

$1 = 1$

This confirms that $\sin 90^{\circ} = 1$.

Exact Value of $\tan 60^{\circ}$

To find the exact value of $\tan 60^{\circ}$, we can use the Quotient Identity.

$\tan 60^{\circ} = \frac{\sin 60^{\circ}}{\cos 60^{\circ}}$

Since $\sin 60^{\circ} = \frac{\sqrt{3}}{2}$ and $\cos 60^{\circ} = \frac{1}{2}$, we can substitute these values in the above equation.

$\tan 60^{\circ} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}$

Simplify:

$\tan 60^{\circ} = \sqrt{3}$

Conclusion

In this article, we have found the exact values of $\cos 45^{\circ}$, $\sin 90^{\circ}$, and $\tan 60^{\circ}$. We used the Pythagorean Identity, Quotient Identity, and Reciprocal Identity to solve these problems. These exact values are essential in trigonometry and are used to solve a wide range of problems related to triangles and waves.

Key Takeaways

• The exact value of $\cos 45^{\circ}$ is $\frac{\sqrt{2}}{2}$.
• The exact value of $\sin 90^{\circ}$ is 1.
• The exact value of $\tan 60^{\circ}$ is $\sqrt{3}$.

Frequently Asked Questions

Q: What is the exact value of $\cos 45^{\circ}$?

A: The exact value of $\cos 45^{\circ}$ is $\frac{\sqrt{2}}{2}$.

Q: What is the exact value of $\sin 90^{\circ}$?

A: The exact value of $\sin 90^{\circ}$ is 1.

Q: What is the exact value of $\tan 60^{\circ}$?

A: The exact value of $\tan 60^{\circ}$ is $\sqrt{3}$.

References

• [1] "Trigonometry" by Michael Corral, 2015.
• [2] "Precalculus" by James Stewart, 2016.
• [3] "Trigonometry for Dummies" by Mary Jane Sterling, 2017.
  Trigonometry Q&A: Frequently Asked Questions =====================================================

Introduction

Trigonometry is a branch of mathematics that deals with the study of triangles, particularly right-angled triangles. It involves the use of trigonometric functions such as sine, cosine, and tangent to solve problems related to the lengths and angles of triangles. In this article, we will answer some frequently asked questions related to trigonometry.

Q&A

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

A: Sine, cosine, and tangent are trigonometric functions that are used to describe the relationships between the angles and sides of a right-angled triangle. The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse, the cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse, and the tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Q: What is the Pythagorean Identity?

A: The Pythagorean Identity is a fundamental identity in trigonometry that states that the sum of the squares of the sine and cosine of an angle is equal to 1. Mathematically, it can be expressed as:

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

Q: How do I find the exact value of a trigonometric function?

A: To find the exact value of a trigonometric function, you can use the Pythagorean Identity, the Quotient Identity, and the Reciprocal Identity. You can also use the unit circle to find the exact values of trigonometric functions.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 that is centered at the origin of a coordinate plane. It is used to find the exact values of trigonometric functions by relating the angles and coordinates of points on the circle.

Q: How do I use the unit circle to find the exact value of a trigonometric function?

A: To use the unit circle to find the exact value of a trigonometric function, you can draw a line from the origin to the point on the circle that corresponds to the angle you are interested in. The coordinates of the point will give you the exact value of the trigonometric function.

Q: What is the difference between a radian and a degree?

A: A radian is a unit of angle that is equal to the angle subtended by an arc of a circle that is equal to the radius of the circle. A degree is a unit of angle that is equal to 1/360 of a full circle.

Q: How do I convert between radians and degrees?

A: To convert between radians and degrees, you can use the following formulas:

• $1$ degree $= \frac{\pi}{180}$ radians
• $1$ radian $= \frac{180}{\pi}$ degrees

Q: What is the sine of a negative angle?

A: The sine of a negative angle is equal to the negative of the sine of the positive angle. Mathematically, it can be expressed as:

$\sin (-x) = -\sin x$

Q: What is the cosine of a negative angle?

A: The cosine of a negative angle is equal to the cosine of the positive angle. Mathematically, it can be expressed as:

$\cos (-x) = \cos x$

Q: What is the tangent of a negative angle?

A: The tangent of a negative angle is equal to the negative of the tangent of the positive angle. Mathematically, it can be expressed as:

$\tan (-x) = -\tan x$

Conclusion

In this article, we have answered some frequently asked questions related to trigonometry. We have covered topics such as the difference between sine, cosine, and tangent, the Pythagorean Identity, the unit circle, and the conversion between radians and degrees. We hope that this article has been helpful in clarifying any doubts you may have had about trigonometry.

Key Takeaways

• The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
• The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
• The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
• The Pythagorean Identity states that the sum of the squares of the sine and cosine of an angle is equal to 1.
• The unit circle is a circle with a radius of 1 that is centered at the origin of a coordinate plane.
• The sine of a negative angle is equal to the negative of the sine of the positive angle.
• The cosine of a negative angle is equal to the cosine of the positive angle.
• The tangent of a negative angle is equal to the negative of the tangent of the positive angle.

References

• [1] "Trigonometry" by Michael Corral, 2015.
• [2] "Precalculus" by James Stewart, 2016.
• [3] "Trigonometry for Dummies" by Mary Jane Sterling, 2017.