Solved on Dec 12, 2023

Find the angle $\theta$ where $180^\circ \leq \theta \leq 360^\circ$ and $\cos \theta = \frac{\sqrt{2}}{2}$ .

STEP 1

Assumptions
1. We are solving for $\theta$ in the interval $180^{\circ} \leq \theta \leq 360^{\circ}$ .
2. The equation to solve is $\cos \theta = \frac{\sqrt{2}}{2}$ .
3. We will use the unit circle and knowledge of trigonometric functions to find the solution.

STEP 2

Recall that $\cos \theta = \frac{\sqrt{2}}{2}$ corresponds to the angles where the x-coordinate of the point on the unit circle is $\frac{\sqrt{2}}{2}$ . These angles are commonly associated with the 45-degree reference angle in the first and fourth quadrants.

STEP 3

However, since we are looking for solutions in the interval $180^{\circ} \leq \theta \leq 360^{\circ}$ , we need to find the angles in the third and fourth quadrants where the cosine function is positive.

STEP 4

The reference angle for the solution in the third and fourth quadrants is $45^{\circ}$ or $\frac{\pi}{4}$ radians.

STEP 5

To find the angle in the third quadrant, we subtract the reference angle from $270^{\circ}$ (or $\frac{3\pi}{2}$ radians), which is the angle corresponding to the negative y-axis on the unit circle.
$\theta_{3rd} = 270^{\circ} - 45^{\circ}$

STEP 6

Calculate the angle in the third quadrant.
$\theta_{3rd} = 225^{\circ}$

STEP 7

To find the angle in the fourth quadrant, we add the reference angle to $270^{\circ}$ (or $\frac{3\pi}{2}$ radians).
$\theta_{4th} = 270^{\circ} + 45^{\circ}$

STEP 8

Calculate the angle in the fourth quadrant.
$\theta_{4th} = 315^{\circ}$

STEP 9

Now we have two possible solutions for $\theta$ within the given interval. These are $\theta = 225^{\circ}$ and $\theta = 315^{\circ}$ .

STEP 10

Verify that both solutions are within the given interval $180^{\circ} \leq \theta \leq 360^{\circ}$ .

STEP 11

Since both $225^{\circ}$ and $315^{\circ}$ are within the interval $180^{\circ} \leq \theta \leq 360^{\circ}$ , they are both valid solutions to the equation.
The solutions to the equation $\cos \theta = \frac{\sqrt{2}}{2}$ for $180^{\circ} \leq \theta \leq 360^{\circ}$ are $\theta = 225^{\circ}$ and $\theta = 315^{\circ}$ .

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Find the angle $\theta$ where $180^\circ \leq \theta \leq 360^\circ$ and $\cos \theta = \frac{\sqrt{2}}{2}$ .

STEP 1

Assumptions
1. We are solving for $\theta$ in the interval $180^{\circ} \leq \theta \leq 360^{\circ}$ .
2. The equation to solve is $\cos \theta = \frac{\sqrt{2}}{2}$ .
3. We will use the unit circle and knowledge of trigonometric functions to find the solution.

STEP 2

Recall that $\cos \theta = \frac{\sqrt{2}}{2}$ corresponds to the angles where the x-coordinate of the point on the unit circle is $\frac{\sqrt{2}}{2}$ . These angles are commonly associated with the 45-degree reference angle in the first and fourth quadrants.

STEP 3

However, since we are looking for solutions in the interval $180^{\circ} \leq \theta \leq 360^{\circ}$ , we need to find the angles in the third and fourth quadrants where the cosine function is positive.

STEP 4

The reference angle for the solution in the third and fourth quadrants is $45^{\circ}$ or $\frac{\pi}{4}$ radians.

STEP 5

To find the angle in the third quadrant, we subtract the reference angle from $270^{\circ}$ (or $\frac{3\pi}{2}$ radians), which is the angle corresponding to the negative y-axis on the unit circle.
$\theta_{3rd} = 270^{\circ} - 45^{\circ}$

STEP 6

Calculate the angle in the third quadrant.
$\theta_{3rd} = 225^{\circ}$

STEP 7

To find the angle in the fourth quadrant, we add the reference angle to $270^{\circ}$ (or $\frac{3\pi}{2}$ radians).
$\theta_{4th} = 270^{\circ} + 45^{\circ}$

STEP 8

Calculate the angle in the fourth quadrant.
$\theta_{4th} = 315^{\circ}$

STEP 9

Now we have two possible solutions for $\theta$ within the given interval. These are $\theta = 225^{\circ}$ and $\theta = 315^{\circ}$ .

STEP 10

Verify that both solutions are within the given interval $180^{\circ} \leq \theta \leq 360^{\circ}$ .

STEP 11

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Find the angle θ\thetaθ where 180∘≤θ≤360∘180^\circ \leq \theta \leq 360^\circ180∘≤θ≤360∘ and cos⁡θ=22\cos \theta = \frac{\sqrt{2}}{2}cosθ=22​​.

STEP 1

STEP 2

Recall that cos⁡θ=22\cos \theta = \frac{\sqrt{2}}{2}cosθ=22​​ corresponds to the angles where the x-coordinate of the point on the unit circle is 22\frac{\sqrt{2}}{2}22​​. These angles are commonly associated with the 45-degree reference angle in the first and fourth quadrants.

STEP 3

However, since we are looking for solutions in the interval 180∘≤θ≤360∘180^{\circ} \leq \theta \leq 360^{\circ}180∘≤θ≤360∘, we need to find the angles in the third and fourth quadrants where the cosine function is positive.

STEP 4

The reference angle for the solution in the third and fourth quadrants is 45∘45^{\circ}45∘ or π4\frac{\pi}{4}4π​ radians.

STEP 5

To find the angle in the third quadrant, we subtract the reference angle from 270∘270^{\circ}270∘ (or 3π2\frac{3\pi}{2}23π​ radians), which is the angle corresponding to the negative y-axis on the unit circle.θ3rd=270∘−45∘\theta_{3rd} = 270^{\circ} - 45^{\circ}θ3rd​=270∘−45∘

STEP 6

Calculate the angle in the third quadrant.θ3rd=225∘\theta_{3rd} = 225^{\circ}θ3rd​=225∘

STEP 7

To find the angle in the fourth quadrant, we add the reference angle to 270∘270^{\circ}270∘ (or 3π2\frac{3\pi}{2}23π​ radians).θ4th=270∘+45∘\theta_{4th} = 270^{\circ} + 45^{\circ}θ4th​=270∘+45∘

STEP 8

Calculate the angle in the fourth quadrant.θ4th=315∘\theta_{4th} = 315^{\circ}θ4th​=315∘

STEP 9

Now we have two possible solutions for θ\thetaθ within the given interval. These are θ=225∘\theta = 225^{\circ}θ=225∘ and θ=315∘\theta = 315^{\circ}θ=315∘.

STEP 10

Verify that both solutions are within the given interval 180∘≤θ≤360∘180^{\circ} \leq \theta \leq 360^{\circ}180∘≤θ≤360∘.

STEP 11

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Find the angle $\theta$ where $180^\circ \leq \theta \leq 360^\circ$ and $\cos \theta = \frac{\sqrt{2}}{2}$ .

Recall that $\cos \theta = \frac{\sqrt{2}}{2}$ corresponds to the angles where the x-coordinate of the point on the unit circle is $\frac{\sqrt{2}}{2}$ . These angles are commonly associated with the 45-degree reference angle in the first and fourth quadrants.

However, since we are looking for solutions in the interval $180^{\circ} \leq \theta \leq 360^{\circ}$ , we need to find the angles in the third and fourth quadrants where the cosine function is positive.

The reference angle for the solution in the third and fourth quadrants is $45^{\circ}$ or $\frac{\pi}{4}$ radians.

To find the angle in the third quadrant, we subtract the reference angle from $270^{\circ}$ (or $\frac{3\pi}{2}$ radians), which is the angle corresponding to the negative y-axis on the unit circle.
$\theta_{3rd} = 270^{\circ} - 45^{\circ}$

Calculate the angle in the third quadrant.
$\theta_{3rd} = 225^{\circ}$

To find the angle in the fourth quadrant, we add the reference angle to $270^{\circ}$ (or $\frac{3\pi}{2}$ radians).
$\theta_{4th} = 270^{\circ} + 45^{\circ}$

Calculate the angle in the fourth quadrant.
$\theta_{4th} = 315^{\circ}$

Now we have two possible solutions for $\theta$ within the given interval. These are $\theta = 225^{\circ}$ and $\theta = 315^{\circ}$ .

Verify that both solutions are within the given interval $180^{\circ} \leq \theta \leq 360^{\circ}$ .