Question

If sin theta= 12/13 and θ is in quadrant II, cos 2 theta= and cos theta = .

Answer 1

Answer:

cos2θ = -0.7041

Cos θ = -0.3847

Step-by-step explanation:

Firstly, we should understand that since θ is in quadrant 2, the value of our cosine will be negative. Only the sine is positive in quadrant 2.

Now the sine of an angle refers to the ratio of the opposite to the adjacent. And since there are three sides to a triangle, we need to find the third side which is the adjacent so that we will be able to evaluate the cosine of the angle.

What to use here is the Pythagoras’ theorem which states that the square of the hypotenuse is equal to the sum of the squares of the adjacent and the opposite.

Since Sine = opposite/hypotenuse, this means that the opposite is 12 and the hypnotist 13

Thus the adjacent let’s say d can be calculated as follows

13^2 = 12^2 + d^2

169 = 144 + d^2

d^2 = 169-144

d^2 = 25

d = √25 = ±5

Since we are on the second quadrant, the value of our adjacent is -5 since the x-coordinate on the second quadrant is negative.(negative x - axis)

The value of cos θ = Adjacent/hypotenuse = -5/13

Cos θ = -5/13

Cos θ = -0.3846

Using trigonometric formulas;

Cos 2θ = cos (θ + θ) = cos θ cos θ - sin θ sin

θ = cos^2 θ - sin^2 θ

Cos 2θ = (-5/13)^2 - (12/13)^2

Cos 2θ = 25/169 - 144/169

Cos 2θ = (25-144)/169 = -119/169

Cos 2θ = -0.7041

Answer 2

There are four quadrants in a coordinate geometry.

The cosine values are $\mathbf{ cos(\theta)=- \frac{5}{13}}$ and $\mathbf{ cos(2\theta) = -\frac{119}{169}}$

The given parameter is:

$\mathbf{sin(\theta) = \frac{12}{13}}$

Using trigonometry ratio, we have:

$\mathbf{sin^2(\theta) + cos^2(\theta)= 1}$

Substitute $\mathbf{sin(\theta) = \frac{12}{13}}$

$\mathbf{(\frac{12}{13})^2 + cos^2(\theta)= 1}$

Expand

$\mathbf{\frac{144}{169} + cos^2(\theta)= 1}$

Collect like terms

$\mathbf{ cos^2(\theta)= 1-\frac{144}{169}}$

Evaluate fraction

$\mathbf{ cos^2(\theta)= \frac{25}{169}}$

Take square roots of both sides

$\mathbf{ cos(\theta)= \pm \frac{5}{13}}$

The angle is in the second quadrant.

So, we have:

$\mathbf{ cos(\theta)=- \frac{5}{13}}$

Also, we have:

$\mathbf{ cos(2\theta) = cos^2(\theta) - sin^2(\theta)}$

So, the equation becomes

$\mathbf{ cos(2\theta) = \frac{25}{169} - \frac{144}{169}}$

Take LCM

$\mathbf{ cos(2\theta) = \frac{25-144}{169}}$

$\mathbf{ cos(2\theta) = -\frac{119}{169}}$

Hence, the values are:

$\mathbf{ cos(\theta)=- \frac{5}{13}}$ and $\mathbf{ cos(2\theta) = -\frac{119}{169}}$

Read more about quadrants at:

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