Solve the following trigonometric and geometry problems: 12. Given (x + 20^) = (2x + 25^), find x and x. 13…

Question

Solve the following trigonometric and geometry problems: 12. Given (x + 20^) = (2x + 25^), find x and x. 13. Given radius r=4 cm and arc length s=5 cm, find the angle theta. 14. In cyclic quadrilatera

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to your problems:

:
Given $\sin(x + 20^\circ) = \cos(2x + 25^\circ)$ .

Use the identity $\cos \theta = \sin(90^\circ - \theta)$ to rewrite the right side of the equation.
$\sin(x + 20^\circ) = \sin(90^\circ - (2x + 25^\circ))$ $\sin(x + 20^\circ) = \sin(90^\circ - 2x - 25^\circ)$ $\sin(x + 20^\circ) = \sin(65^\circ - 2x)$
Equate the angles since the sine values are equal.
$x + 20^\circ = 65^\circ - 2x$
Solve for $x$ .
$x + 2x = 65^\circ - 20^\circ$ $3x = 45^\circ$ $x = \frac{45^\circ}{3}$ $x = 15^\circ$
Find the value of $\tan x$ .
$\tan x = \tan(15^\circ)$ $\tan(15^\circ) \approx 0.2679$
The value of $x$ is $\boxed{15^\circ}$ and the value of $\tan x$ is $\boxed{0.2679}$ .

:
Given radius $r = 4$ cm and arc length $s = 5$ cm.
Use the arc length formula $s = r\theta$ , where $\theta$ is in radians.
$5 \text{ cm} = (4 \text{ cm}) \theta$
Solve for $\theta$ .
$\theta = \frac{5 \text{ cm}}{4 \text{ cm}}$ $\theta = 1.25 \text{ radians}$
The angle subtended by the arc at the center is $\boxed{1.25 \text{ radians}}$ .

:
The figure shows a cyclic quadrilateral ABCD.
In a cyclic quadrilateral, opposite angles sum to $180^\circ$ .
Angles D and B are opposite.
$\angle D + \angle B = 180^\circ$
Substitute the given expressions for $\angle D$ and $\angle B$ .
$120^\circ + 6x^\circ = 180^\circ$
Solve the equation for $x$ .
$6x^\circ = 180^\circ - 120^\circ$ $6x^\circ = 60^\circ$ $x = \frac{60}{6}$ $x = 10$
The value of $x$ is $\boxed{10}$ .

Solve the following trigonometric and geometry problems: 12. Given (x + 20^) = (2x + 25^), find x and x. 13. Given radius r=4 cm and arc length s=5 cm, find the angle theta. 14. In cyclic quadrilatera

Step-by-step solution

Use the identity $\cos \theta = \sin(90^\circ - \theta)$ to rewrite the right side of the equation.

Equate the angles since the sine values are equal.

Solve for $x$ .

Find the value of $\tan x$ .

Use the arc length formula $s = r\theta$ , where $\theta$ is in radians.

Solve for $\theta$ .

In a cyclic quadrilateral, opposite angles sum to $180^\circ$ .

Substitute the given expressions for $\angle D$ and $\angle B$ .

Solve the equation for $x$ .

Still stuck on this one?

More Mathematics Questions

Solve the following trigonometric and geometry problems: 12. Given (x + 20^) = (2x + 25^), find x and x. 13. Given radius r=4 cm and arc length s=5 cm, find the angle theta. 14. In cyclic quadrilatera

Step-by-step solution

Use the identity cos⁡θ=sin⁡(90∘−θ)\cos \theta = \sin(90^\circ - \theta)cosθ=sin(90∘−θ) to rewrite the right side of the equation.

Equate the angles since the sine values are equal.

Solve for xxx.

Find the value of tan⁡x\tan xtanx.

Use the arc length formula s=rθs = r\thetas=rθ, where θ\thetaθ is in radians.

Solve for θ\thetaθ.

In a cyclic quadrilateral, opposite angles sum to 180∘180^\circ180∘.

Substitute the given expressions for ∠D\angle D∠D and ∠B\angle B∠B.

Solve the equation for xxx.

Still stuck on this one?

More Mathematics Questions

Use the identity $\cos \theta = \sin(90^\circ - \theta)$ to rewrite the right side of the equation.

Solve for $x$ .

Find the value of $\tan x$ .

Use the arc length formula $s = r\theta$ , where $\theta$ is in radians.

Solve for $\theta$ .

In a cyclic quadrilateral, opposite angles sum to $180^\circ$ .

Substitute the given expressions for $\angle D$ and $\angle B$ .

Solve the equation for $x$ .