Given tan ⁡ θ = 5 12 \tan\theta=\frac{5}{12} tan θ = 12 5 and 0 < θ \theta θ < π 2 \frac{\pi}{2} 2 π , find cos ⁡ ( 2 θ ) \cos\left(2\theta\right) cos ( 2 θ ) .

119 169 \frac{119}{169} 169 119

− 199 169 -\frac{199}{169} − 169 199

144 169 \frac{144}{169} 169 144

6. Trigonometric Identities and More Equations

Double Angle Identities

6. Trigonometric Identities and More Equations

Double Angle Identities: Videos & Practice Problems

Video Lessons Practice Worksheet

Topic summary

Double angle identities are derived from sum formulas and simplify trigonometric expressions. Key identities include: sin2(θ)=2⁢sin(θ)⁢cos(θ), cos2(θ)=cos2(θ)-sin2(θ), and tan2(θ)=2tan(θ)/(1-tan2(θ)). Recognizing parts of these identities aids in simplifying expressions effectively, enhancing problem-solving skills in trigonometry.

concept

Double Angle Identities

Video duration:

Double Angle Identities Video Summary

Trigonometric identities are essential tools in simplifying and solving trigonometric expressions. Among these identities, double angle identities are particularly useful, derived from the sum formulas for sine, cosine, and tangent when the same angle is used twice. Understanding these identities allows for more efficient problem-solving in trigonometry.

The double angle identity for sine states that:

$$\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$$

For cosine, the double angle identity can be expressed in three forms:

$$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$$

Alternatively, using the Pythagorean identity, it can also be written as:

$$\cos(2\theta) = 2\cos^2(\theta) - 1$$

$$\cos(2\theta) = 1 - 2\sin^2(\theta)$$

For tangent, the double angle identity is given by:

$$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$$

These identities are particularly useful when the argument of a trigonometric function is expressed as twice an angle. For example, if you encounter an expression like $\cos^2(\frac{\pi}{12}) - \sin^2(\frac{\pi}{12})$, you can recognize it as the cosine double angle identity, simplifying it to:

$$\cos\left(2 \cdot \frac{\pi}{12}\right) = \cos\left(\frac{\pi}{6}\right)$$

Knowing that $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$ allows for quick evaluation.

When simplifying expressions, always look for parts of these identities. For instance, if you have $\sin(15^\circ) \cos(15^\circ)$, you can identify it as part of the sine double angle identity. By manipulating the identity:

$$\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$$

you can rewrite it as:

$$\frac{\sin(30^\circ)}{2}$$

Thus, recognizing and applying these identities can significantly streamline the process of simplifying trigonometric expressions.

As you continue to learn more identities, practice is key. The more familiar you become with these relationships, the easier it will be to identify when and how to apply them in various problems.

example

Example 1

Video duration:

Example 1 Video Summary

In this problem, we are tasked with verifying the identity that states the sine of $2\theta$ over the tangent of $\theta$ is equal to the cosine of $2\theta$ plus 1. To approach this verification, we start with the more complex side, which in this case is the left side containing a fraction.

We begin by applying the double angle formula for sine, which states that:

\[\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\]

Thus, we can rewrite the left side as:

\[\frac{\sin(2\theta)}{\tan(\theta)} = \frac{2 \sin(\theta) \cos(\theta)}{\tan(\theta)}\]

Next, we express the tangent function in terms of sine and cosine:

\[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]

Substituting this into our expression gives us:

\[\frac{2 \sin(\theta) \cos(\theta)}{\frac{\sin(\theta)}{\cos(\theta)}}\]

When dividing by a fraction, we multiply by its reciprocal:

\[2 \sin(\theta) \cos(\theta) \cdot \frac{\cos(\theta)}{\sin(\theta)} = 2 \cos^2(\theta)\]

Now, we need to relate $2 \cos^2(\theta)$ to the right side of the identity, which is $\cos(2\theta) + 1$. We recall the cosine double angle formula:

\[\cos(2\theta) = 2 \cos^2(\theta) - 1\]

Rearranging this formula gives us:

\[2 \cos^2(\theta) = \cos(2\theta) + 1\]

Thus, we can substitute $2 \cos^2(\theta)$ back into our expression:

\[2 \cos^2(\theta) = \cos(2\theta) + 1\]

Now we see that both sides of the original identity are equal:

\[\frac{\sin(2\theta)}{\tan(\theta)} = \cos(2\theta) + 1\]

Therefore, we have successfully verified the identity. If you have any questions or need further clarification, feel free to ask!

example

Example 2

Video duration:

Example 2 Video Summary

In this problem, we are tasked with verifying the identity that states the cotangent of $2\theta$ is equal to the cotangent of $\theta$ minus $\frac{1}{2} \cos \theta \sin \theta$. To begin, we focus on the more complex side of the equation, which is the right side. The cotangent function can be expressed in terms of sine and cosine, specifically, $\cot \theta = \frac{\cos \theta}{\sin \theta}$. By substituting this into our equation, we rewrite the right side as:

\[\frac{\cos \theta}{\sin \theta} - \frac{1}{2} \cos \theta \sin \theta\]

Next, we need to combine these fractions. The common denominator for the two fractions is $2 \cos \theta \sin \theta$. We multiply the first fraction by $\frac{2 \cos \theta}{2 \cos \theta}$ to achieve this common denominator:

\[\frac{2 \cos^2 \theta}{2 \sin \theta \cos \theta} - \frac{1}{2} \cos \theta \sin \theta = \frac{2 \cos^2 \theta - 1}{2 \sin \theta \cos \theta}\]

Now, we can identify a trigonometric identity in the numerator. The expression $2 \cos^2 \theta - 1$ corresponds to the double angle identity for cosine, which states that $\cos(2\theta) = 2 \cos^2 \theta - 1$. Thus, we can replace the numerator with $\cos(2\theta)$:

\[\frac{\cos(2\theta)}{2 \sin \theta \cos \theta}\]

In the denominator, we recognize that $2 \sin \theta \cos \theta$ is equivalent to $\sin(2\theta)$ based on the double angle identity for sine. Therefore, we can rewrite the right side as:

\[\frac{\cos(2\theta)}{\sin(2\theta)}\]

This expression is precisely the definition of cotangent, leading us to:

\[\cot(2\theta)\]

Thus, we have successfully verified the identity, confirming that both sides of the equation are indeed equal. This process illustrates the importance of recognizing and applying trigonometric identities to simplify expressions effectively.

example

Example 3

Video duration:

Example 3 Video Summary

In trigonometry, when tasked with finding a function of double angles, such as $ \sin(2\theta) $, it is essential to utilize double angle identities effectively. For this specific case, we know that $ \cos(\theta) = \frac{4}{5} $ and that $ \theta $ is in the first quadrant, where both sine and cosine are positive.

To find $ \sin(2\theta) $, we can apply the double angle identity:

$ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) $.

First, we need to determine $ \sin(\theta) $. Given $ \cos(\theta) = \frac{4}{5} $, we can visualize this using a right triangle where the adjacent side is 4 and the hypotenuse is 5. To find the opposite side, we can use the Pythagorean theorem:

$ a^2 + b^2 = c^2 $, where $ a $ is the opposite side, $ b $ is the adjacent side, and $ c $ is the hypotenuse. Here, we have:

$ 4^2 + a^2 = 5^2 $

$ 16 + a^2 = 25 $

$ a^2 = 9 $

$ a = 3 $.

Thus, the opposite side measures 3. Now we can find $ \sin(\theta) $:

$ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} $.

With both $ \sin(\theta) $ and $ \cos(\theta) $ known, we can substitute these values into the double angle identity:

$ \sin(2\theta) = 2 \left(\frac{3}{5}\right) \left(\frac{4}{5}\right) = 2 \cdot \frac{12}{25} = \frac{24}{25} $.

Therefore, the final result for $ \sin(2\theta) $ is $ \frac{24}{25} $. This process illustrates the importance of understanding trigonometric identities and the relationships between the sides of a triangle in solving problems involving angles and their functions.

Problem

Given $\tan\theta=\frac{5}{12}$ and 0 < $\theta$ < $\frac{\pi}{2}$ , find $\cos\left(2\theta\right)$ .

$-\frac{199}{169}$

$\frac{119}{169}$

$\frac{144}{169}$

Do you want more practice?

We have more practice problems on Double Angle Identities

Here’s what students ask on this topic:

What are the double angle identities in trigonometry?

Double angle identities are trigonometric identities that express trigonometric functions of double angles (2θ) in terms of functions of the single angle (θ). The key double angle identities are:

$\sin^{2} (θ) = 2 \sin (θ) \cos (θ)$

$\cos^{2} (θ) = \cos^{2} (θ) - \sin^{2} (θ)$

$\tan^{2} (θ) = \frac{2 \tan (θ)}{1 - \tan^{2} (θ)}$

How do you derive the double angle identity for sine?

The double angle identity for sine is derived from the sum formula for sine. The sum formula states:

$\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$

By setting A = θ and B = θ, we get:

$\sin (2 θ) = \sin (θ) \cos (θ) + \cos (θ) \sin (θ)$

Since the terms are identical, we can combine them:

$\sin (2 θ) = 2 \sin (θ) \cos (θ)$

When should you use double angle identities in trigonometry problems?

Double angle identities should be used in trigonometry problems whenever the argument of the trigonometric function is 2θ. Additionally, they are useful when you recognize parts of the identity within an expression. For example, if you see an expression like 2sin(θ)cos(θ), you can use the double angle identity for sine to simplify it to sin(2θ). Recognizing these patterns helps in simplifying complex trigonometric expressions and solving equations more efficiently.

How can you simplify the expression sin(15°)cos(15°) using double angle identities?

To simplify sin(15°)cos(15°) using double angle identities, recognize that it resembles part of the double angle identity for sine:

$\sin (2 θ) = 2 \sin (θ) \cos (θ)$

Divide both sides by 2:

$\frac{\sin}{(}$

Set θ = 15°:

$\frac{\sin}{(}$

So, sin(15°)cos(15°) simplifies to sin(30°)/2, which is 1/2 * 1/2 = 1/4.

What are the alternate forms of the cosine double angle identity?

The cosine double angle identity has three forms, derived using Pythagorean identities:

$\cos^{2} (θ) = \cos^{2} (θ) - \sin^{2} (θ)$

$\cos^{2} (θ) = 2 \cos^{2} (θ) - 1$

$\cos^{2} (θ) = 1 - 2 \sin^{2} (θ)$

These forms are useful in different scenarios for simplifying trigonometric expressions.

Your Trigonometry tutors

Patrick Ford

Physics and Math Lead Instructor

Double Angle Identities: Videos & Practice Problems

Double Angle Identities

Double Angle Identities Video Summary

Example 1

Example 1 Video Summary

Example 2

Example 2 Video Summary

Example 3

Example 3 Video Summary

Given tanθ=512\tan\theta=\frac{5}{12}tanθ=125 and 0 < θ\thetaθ < π2\frac{\pi}{2}2π, find cos(2θ)\cos\left(2\theta\right)cos(2θ).

Do you want more practice?

Here’s what students ask on this topic:

What are the double angle identities in trigonometry?

How do you derive the double angle identity for sine?

When should you use double angle identities in trigonometry problems?

How can you simplify the expression sin(15°)cos(15°) using double angle identities?

What are the alternate forms of the cosine double angle identity?

Your Trigonometry tutors

Given $\tan\theta=\frac{5}{12}$ and 0 < $\theta$ < $\frac{\pi}{2}$ , find $\cos\left(2\theta\right)$ .