QuestionFind $\tan \theta$ for the point $P(5,4)$ on the circle $x^{2}+y^{2}=r^{2}$ .

Studdy Solution

STEP 1

Assumptions1. The point is on the circle $x^{}+y^{}=r^{}$ and also on the terminal side of an angle $\theta$ in standard position. . The coordinates of point are (5,4).
3. We are asked to find the value of $\tan \theta$ .

STEP 2

In the unit circle, the x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle. Therefore, we can write $\cos \theta = \frac{x}{r}$ $\sin \theta = \frac{y}{r}$

STEP 3

The tangent of an angle is the ratio of the sine to the cosine of that angle. Therefore, we can write $\tan \theta = \frac{\sin \theta}{\cos \theta}$

STEP 4

Substitute the expressions for $\sin \theta$ and $\cos \theta$ from2 into the equation from3 to get $\tan \theta = \frac{y/x}{r/r}$

STEP 5

implify the equation from4 to get $\tan \theta = \frac{y}{x}$

STEP 6

Substitute the given coordinates of point into the equation from5 to find the value of $\tan \theta$ $\tan \theta = \frac{4}{5}$ Therefore, the value of $\tan \theta$ is $\frac{4}{5}$ .

Was this helpful?

QuestionFind $\tan \theta$ for the point $P(5,4)$ on the circle $x^{2}+y^{2}=r^{2}$ .

Studdy Solution

STEP 1

Assumptions1. The point is on the circle $x^{}+y^{}=r^{}$ and also on the terminal side of an angle $\theta$ in standard position. . The coordinates of point are (5,4).
3. We are asked to find the value of $\tan \theta$ .

STEP 2

In the unit circle, the x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle. Therefore, we can write $\cos \theta = \frac{x}{r}$ $\sin \theta = \frac{y}{r}$

STEP 3

The tangent of an angle is the ratio of the sine to the cosine of that angle. Therefore, we can write $\tan \theta = \frac{\sin \theta}{\cos \theta}$

STEP 4

Substitute the expressions for $\sin \theta$ and $\cos \theta$ from2 into the equation from3 to get $\tan \theta = \frac{y/x}{r/r}$

STEP 5

implify the equation from4 to get $\tan \theta = \frac{y}{x}$

STEP 6

Substitute the given coordinates of point into the equation from5 to find the value of $\tan \theta$ $\tan \theta = \frac{4}{5}$ Therefore, the value of $\tan \theta$ is $\frac{4}{5}$ .

Get Studdy

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.

QuestionFind tan⁡θ\tan \thetatanθ for the point P(5,4)P(5,4)P(5,4) on the circle x2+y2=r2x^{2}+y^{2}=r^{2}x2+y2=r2.

Studdy Solution

STEP 1

Assumptions1. The point is on the circle x+y=rx^{}+y^{}=r^{}x+y=r and also on the terminal side of an angle θ\thetaθ in standard position. . The coordinates of point are (5,4).3. We are asked to find the value of tan⁡θ\tan \thetatanθ.

STEP 2

In the unit circle, the x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle. Therefore, we can writecos⁡θ=xr\cos \theta = \frac{x}{r}cosθ=rx​sin⁡θ=yr\sin \theta = \frac{y}{r}sinθ=ry​

STEP 3

The tangent of an angle is the ratio of the sine to the cosine of that angle. Therefore, we can writetan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ​

STEP 4

Substitute the expressions for sin⁡θ\sin \thetasinθ and cos⁡θ\cos \thetacosθ from2 into the equation from3 to gettan⁡θ=y/xr/r\tan \theta = \frac{y/x}{r/r}tanθ=r/ry/x​

STEP 5

implify the equation from4 to gettan⁡θ=yx\tan \theta = \frac{y}{x}tanθ=xy​

STEP 6

Substitute the given coordinates of point into the equation from5 to find the value of tan⁡θ\tan \thetatanθtan⁡θ=45\tan \theta = \frac{4}{5}tanθ=54​Therefore, the value of tan⁡θ\tan \thetatanθ is 45\frac{4}{5}54​.

Get Studdy

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.

QuestionFind $\tan \theta$ for the point $P(5,4)$ on the circle $x^{2}+y^{2}=r^{2}$ .

Assumptions1. The point is on the circle $x^{}+y^{}=r^{}$ and also on the terminal side of an angle $\theta$ in standard position. . The coordinates of point are (5,4).
3. We are asked to find the value of $\tan \theta$ .

In the unit circle, the x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle. Therefore, we can write $\cos \theta = \frac{x}{r}$ $\sin \theta = \frac{y}{r}$

The tangent of an angle is the ratio of the sine to the cosine of that angle. Therefore, we can write $\tan \theta = \frac{\sin \theta}{\cos \theta}$

Substitute the expressions for $\sin \theta$ and $\cos \theta$ from2 into the equation from3 to get $\tan \theta = \frac{y/x}{r/r}$

implify the equation from4 to get $\tan \theta = \frac{y}{x}$

Substitute the given coordinates of point into the equation from5 to find the value of $\tan \theta$ $\tan \theta = \frac{4}{5}$ Therefore, the value of $\tan \theta$ is $\frac{4}{5}$ .