Math / Trigonometry

QuestionWhich of the following trigonometric expressions are equivalent? Explain your answer. $\cos ^{2} \theta, \cos 2 \theta, \cos (\theta)^{2}, \cos \theta \cdot \cos \theta, \cos \left(\theta^{2}\right)$

Studdy Solution

STEP 1

1. We are given several trigonometric expressions involving cosine and need to determine which are equivalent.
2. Equivalent expressions will yield the same value for any angle $\theta$ .
3. We will use trigonometric identities and algebraic simplification to compare the expressions.

STEP 2

1. Analyze each expression individually.
2. Simplify or rewrite expressions using trigonometric identities.
3. Compare simplified expressions to determine equivalence.

STEP 3

Analyze the expression $\cos^2 \theta$ :
This expression represents $(\cos \theta)^2$ , which is the square of the cosine of $\theta$ .

STEP 4

Analyze the expression $\cos 2\theta$ :
This expression represents the cosine of the angle $2\theta$ . Using the double angle identity, we have:
$\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1$

STEP 5

Analyze the expression $\cos(\theta)^2$ :
This expression is equivalent to $\cos^2 \theta$ because it represents the square of the cosine of $\theta$ .

STEP 6

Analyze the expression $\cos \theta \cdot \cos \theta$ :
This expression is equivalent to $(\cos \theta)^2$ , which is $\cos^2 \theta$ .

STEP 7

Analyze the expression $\cos(\theta^2)$ :
This expression represents the cosine of the square of $\theta$ , which is different from the other expressions as it involves $\theta^2$ inside the cosine function.

STEP 8

Compare the expressions:
- $\cos^2 \theta$ , $\cos(\theta)^2$ , and $\cos \theta \cdot \cos \theta$ are all equivalent because they represent the square of the cosine of $\theta$ . - $\cos 2\theta$ is not equivalent to the above because it involves a double angle identity. - $\cos(\theta^2)$ is not equivalent to any of the others because it involves the cosine of $\theta^2$ .
The equivalent expressions are $\cos^2 \theta$ , $\cos(\theta)^2$ , and $\cos \theta \cdot \cos \theta$ .

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QuestionWhich of the following trigonometric expressions are equivalent? Explain your answer. $\cos ^{2} \theta, \cos 2 \theta, \cos (\theta)^{2}, \cos \theta \cdot \cos \theta, \cos \left(\theta^{2}\right)$

Studdy Solution

STEP 1

1. We are given several trigonometric expressions involving cosine and need to determine which are equivalent.
2. Equivalent expressions will yield the same value for any angle $\theta$ .
3. We will use trigonometric identities and algebraic simplification to compare the expressions.

STEP 2

1. Analyze each expression individually.
2. Simplify or rewrite expressions using trigonometric identities.
3. Compare simplified expressions to determine equivalence.

STEP 3

Analyze the expression $\cos^2 \theta$ :
This expression represents $(\cos \theta)^2$ , which is the square of the cosine of $\theta$ .

STEP 4

Analyze the expression $\cos 2\theta$ :
This expression represents the cosine of the angle $2\theta$ . Using the double angle identity, we have:
$\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1$

STEP 5

Analyze the expression $\cos(\theta)^2$ :
This expression is equivalent to $\cos^2 \theta$ because it represents the square of the cosine of $\theta$ .

STEP 6

Analyze the expression $\cos \theta \cdot \cos \theta$ :
This expression is equivalent to $(\cos \theta)^2$ , which is $\cos^2 \theta$ .

STEP 7

Analyze the expression $\cos(\theta^2)$ :
This expression represents the cosine of the square of $\theta$ , which is different from the other expressions as it involves $\theta^2$ inside the cosine function.

STEP 8

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Studdy Solution

STEP 1

1. We are given several trigonometric expressions involving cosine and need to determine which are equivalent.2. Equivalent expressions will yield the same value for any angle θ\thetaθ.3. We will use trigonometric identities and algebraic simplification to compare the expressions.

STEP 2

1. Analyze each expression individually.2. Simplify or rewrite expressions using trigonometric identities.3. Compare simplified expressions to determine equivalence.

STEP 3

Analyze the expression cos⁡2θ\cos^2 \thetacos2θ:This expression represents (cos⁡θ)2(\cos \theta)^2(cosθ)2, which is the square of the cosine of θ\thetaθ.

STEP 4

STEP 5

Analyze the expression cos⁡(θ)2\cos(\theta)^2cos(θ)2:This expression is equivalent to cos⁡2θ\cos^2 \thetacos2θ because it represents the square of the cosine of θ\thetaθ.

STEP 6

Analyze the expression cos⁡θ⋅cos⁡θ\cos \theta \cdot \cos \thetacosθ⋅cosθ:This expression is equivalent to (cos⁡θ)2(\cos \theta)^2(cosθ)2, which is cos⁡2θ\cos^2 \thetacos2θ.

STEP 7

Analyze the expression cos⁡(θ2)\cos(\theta^2)cos(θ2):This expression represents the cosine of the square of θ\thetaθ, which is different from the other expressions as it involves θ2\theta^2θ2 inside the cosine function.

STEP 8

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Studdy solves anything!

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. We are given several trigonometric expressions involving cosine and need to determine which are equivalent.
2. Equivalent expressions will yield the same value for any angle $\theta$ .
3. We will use trigonometric identities and algebraic simplification to compare the expressions.

1. Analyze each expression individually.
2. Simplify or rewrite expressions using trigonometric identities.
3. Compare simplified expressions to determine equivalence.

Analyze the expression $\cos^2 \theta$ :
This expression represents $(\cos \theta)^2$ , which is the square of the cosine of $\theta$ .

Analyze the expression $\cos(\theta)^2$ :
This expression is equivalent to $\cos^2 \theta$ because it represents the square of the cosine of $\theta$ .

Analyze the expression $\cos \theta \cdot \cos \theta$ :
This expression is equivalent to $(\cos \theta)^2$ , which is $\cos^2 \theta$ .

Analyze the expression $\cos(\theta^2)$ :
This expression represents the cosine of the square of $\theta$ , which is different from the other expressions as it involves $\theta^2$ inside the cosine function.