QuestionProve that $\operatorname{Cos}^{2} \theta+\operatorname{Sin}^{2} \theta=1$ . Do all rough work on the answer scripts.

Studdy Solution

STEP 1

Assumptions1. $\theta$ is a real number. . We are using standard trigonometric identities and properties.
3. We are assuming the standard definitions of sine and cosine functions.

STEP 2

We will use the Pythagorean identity in trigonometry, which states that for any real number $\theta$ , the square of the sine of $\theta$ plus the square of the cosine of $\theta$ equals1.
$\operatorname{Sin}^{2} \theta + \operatorname{Cos}^{2} \theta =1$

STEP 3

This is a fundamental identity in trigonometry and does not require further proof. However, if we want to derive it, we can do so from the unit circle definition of sine and cosine.

STEP 4

In the unit circle, for any angle $\theta$ , the cosine of $\theta$ is the x-coordinate of the point on the unit circle, and the sine of $\theta$ is the y-coordinate.

STEP 5

The equation of the unit circle is $x^{2} + y^{2} =1$ .

STEP 6

Substitute $cos \theta$ for $x$ and $sin \theta$ for $y$ in the equation of the unit circle.
$\operatorname{Cos}^{2} \theta + \operatorname{Sin}^{2} \theta =1$

STEP 7

This is exactly the identity we wanted to prove. Therefore, we have shown that for any real number $\theta$ , $\operatorname{Cos}^{2} \theta + \operatorname{Sin}^{2} \theta =1$ .

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QuestionProve that $\operatorname{Cos}^{2} \theta+\operatorname{Sin}^{2} \theta=1$ . Do all rough work on the answer scripts.

Studdy Solution

STEP 1

Assumptions1. $\theta$ is a real number. . We are using standard trigonometric identities and properties.
3. We are assuming the standard definitions of sine and cosine functions.

STEP 2

We will use the Pythagorean identity in trigonometry, which states that for any real number $\theta$ , the square of the sine of $\theta$ plus the square of the cosine of $\theta$ equals1.
$\operatorname{Sin}^{2} \theta + \operatorname{Cos}^{2} \theta =1$

STEP 3

This is a fundamental identity in trigonometry and does not require further proof. However, if we want to derive it, we can do so from the unit circle definition of sine and cosine.

STEP 4

In the unit circle, for any angle $\theta$ , the cosine of $\theta$ is the x-coordinate of the point on the unit circle, and the sine of $\theta$ is the y-coordinate.

STEP 5

The equation of the unit circle is $x^{2} + y^{2} =1$ .

STEP 6

Substitute $cos \theta$ for $x$ and $sin \theta$ for $y$ in the equation of the unit circle.
$\operatorname{Cos}^{2} \theta + \operatorname{Sin}^{2} \theta =1$

STEP 7

This is exactly the identity we wanted to prove. Therefore, we have shown that for any real number $\theta$ , $\operatorname{Cos}^{2} \theta + \operatorname{Sin}^{2} \theta =1$ .

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QuestionProve that Cos⁡2θ+Sin⁡2θ=1\operatorname{Cos}^{2} \theta+\operatorname{Sin}^{2} \theta=1Cos2θ+Sin2θ=1. Do all rough work on the answer scripts.

Studdy Solution

STEP 1

Assumptions1. θ\thetaθ is a real number. . We are using standard trigonometric identities and properties.3. We are assuming the standard definitions of sine and cosine functions.

STEP 2

We will use the Pythagorean identity in trigonometry, which states that for any real number θ\thetaθ, the square of the sine of θ\thetaθ plus the square of the cosine of θ\thetaθ equals1.Sin⁡2θ+Cos⁡2θ=1\operatorname{Sin}^{2} \theta + \operatorname{Cos}^{2} \theta =1Sin2θ+Cos2θ=1

STEP 3

This is a fundamental identity in trigonometry and does not require further proof. However, if we want to derive it, we can do so from the unit circle definition of sine and cosine.

STEP 4

In the unit circle, for any angle θ\thetaθ, the cosine of θ\thetaθ is the x-coordinate of the point on the unit circle, and the sine of θ\thetaθ is the y-coordinate.

STEP 5

The equation of the unit circle is x2+y2=1x^{2} + y^{2} =1x2+y2=1.

STEP 6

Substitute cosθcos \thetacosθ for xxx and sinθsin \thetasinθ for yyy in the equation of the unit circle.Cos⁡2θ+Sin⁡2θ=1\operatorname{Cos}^{2} \theta + \operatorname{Sin}^{2} \theta =1Cos2θ+Sin2θ=1

STEP 7

This is exactly the identity we wanted to prove. Therefore, we have shown that for any real number θ\thetaθ, Cos⁡2θ+Sin⁡2θ=1\operatorname{Cos}^{2} \theta + \operatorname{Sin}^{2} \theta =1Cos2θ+Sin2θ=1.

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Studdy solves anything!

Start learning now

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

QuestionProve that $\operatorname{Cos}^{2} \theta+\operatorname{Sin}^{2} \theta=1$ . Do all rough work on the answer scripts.

Assumptions1. $\theta$ is a real number. . We are using standard trigonometric identities and properties.
3. We are assuming the standard definitions of sine and cosine functions.

We will use the Pythagorean identity in trigonometry, which states that for any real number $\theta$ , the square of the sine of $\theta$ plus the square of the cosine of $\theta$ equals1.
$\operatorname{Sin}^{2} \theta + \operatorname{Cos}^{2} \theta =1$

In the unit circle, for any angle $\theta$ , the cosine of $\theta$ is the x-coordinate of the point on the unit circle, and the sine of $\theta$ is the y-coordinate.

The equation of the unit circle is $x^{2} + y^{2} =1$ .

Substitute $cos \theta$ for $x$ and $sin \theta$ for $y$ in the equation of the unit circle.
$\operatorname{Cos}^{2} \theta + \operatorname{Sin}^{2} \theta =1$

This is exactly the identity we wanted to prove. Therefore, we have shown that for any real number $\theta$ , $\operatorname{Cos}^{2} \theta + \operatorname{Sin}^{2} \theta =1$ .