What value of x makes this equation true?

If sin theta = 2/3 and sex theta < 0 , find cos theta and tan theta

FromTheMoon [43]

Answer:

$\displaystyle cos\theta=-\frac{\sqrt{5}}{3}$

$\displaystyle tan\theta=-\frac{2\sqrt{5}}{5}$

Step-by-step explanation:

<u>Trigonometric Formulas</u>

To solve this problem, we must recall some basic relations and concepts.

The main trigonometric identity relates the sine to the cosine:

$sin^2\theta+cos^2\theta=1$

The tangent can be found by

$\displaystyle tan\theta=\frac{sin\theta}{cos\theta}$

The cosine and the secant are related by

$\displaystyle cos\theta=\frac{1}{sec\theta}$

They both have the same sign.

The sine is positive in the first and second quadrants, the cosine is positive in the first and fourth quadrants.

The sine is negative in the third and fourth quadrants, the cosine is negative in the second and third quadrants.

We are given

$\displaystyle sin\theta=\frac{2}{3}$

Find the cosine by solving

$sin^2\theta+cos^2\theta=1$

$\displaystyle \left(\frac{2}{3}\right)^2+cos^2\theta=1$

$\displaystyle cos^2\theta=1-\left(\frac{2}{3}\right)^2=1-\frac{4}{9}=\frac{5}{9}$

$\displaystyle cos\theta=\sqrt{\frac{5}{9}}=-\frac{\sqrt{5}}{3}$

$\boxed{\displaystyle cos\theta=-\frac{\sqrt{5}}{3}}$

We have placed the negative sign because we know the secant ('sex') is negative and they both have the same sign.

Now compute the tangent

$\displaystyle tan\theta=\frac{sin\theta}{cos\theta}=\frac{\frac{2}{3}}{-\frac{\sqrt{5}}{3}}=-\frac{2}{\sqrt{5}}$

Rationalizing

$\displaystyle tan\theta=-\frac{2}{\sqrt{5}}=-\frac{2\sqrt{5}}{5}$

$\boxed{\displaystyle tan\theta=-\frac{2\sqrt{5}}{5}}$

5 0

4 years ago

Lana draws a triangle( LMN) on the coordinate plane. What is the perimeter of the triangle( LMN)? Round to the nearest unit.

Dmitriy789 [7]

Answer:

25

Step-by-step explanation:

The longest side is 12.

The two other sides of the triangle= $\sqrt{3 squared+6 squared}$ =(9+36)square rooted=45 square rooted is about 6.7.

6.7*2=13.4

13.4+12=25.4

25.4 rounded to the nearest unit is 25.

8 0

4 years ago

A trapezoid was broken into a rectangle and a triangle. A trapezoid is broken into a rectangle and a triangle. The rectangle has

goldenfox [79]

Answer:

b = 2 cm.

Step-by-step explanation:

A trapezoid is broken into a rectangle and a triangle. So, it is a right trapezoid.

The rectangle has a base of 8 centimeters and a height of 6 centimeters. The triangle has a base of 2 centimeters and a height of 6 centimeters.

Now, we are asked the length, b, of the base of the triangle.

It is clear from the given information that the base of the triangle is, b = 2 cm. (Answer)

6 0

3 years ago

Read 2 more answers

Executives at a large multinational company are being accused of spending too much time on email and not enough time with their

Elena L [17]

Answer:

We conclude that there is not enough evidence to support the claim that company is spending too much time on email.

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = 96 minutes

Sample mean, $\bar{x}$ = 91 minutes

Sample size, n = 100

Alpha, α = 0.05

Sample standard deviation, s = 25 minutes

First, we design the null and the alternate hypothesis

$H_{0}: \mu \leq 96\text{ minutes}\\H_A: \mu > 96\text{ minutes}$

We use one-tailed t test to perform this hypothesis.

Formula:

$t_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}} }$

Putting all the values, we have

$t_{stat} = \displaystyle\frac{91 - 96}{\frac{25}{\sqrt{100}} } = -2$

Now, $t_{critical} \text{ at 0.05 level of significance, 99 degree of freedom } = 1.6603$

Since,

The calculated test statistic is less than the critical value, we fail to reject the null hypothesis and accept it.

Thus, we conclude that there is not enough evidence to support the claim that company is spending too much time on email.

6 0

4 years ago

2x+5y=11 and 4x+3y=1 solve with elimination

storchak [24]

Multiply you first equation by 2 to get 4x + 10y = 22
Now the x terms can be eliminated and cancelled out using elimination.
4x + 10y = 22
4x + 3y = 1

To eliminate you need to "subtract", so you need to multiply one of the equations by -1.

4x + 10y = 22
-4x -3y = -1
------------------
0 + 7y = 21
y = 3

Now plug 3 into either one of the equations to get x.

7 0

4 years ago