Homework

Find the values of angles θ that satisfy cos(2θ) + sin(3θ) = 1 within the range [0, 2π]

First, we rewrite the given equation: $$ \cos(2\theta) + \sin(3\theta) = 1 $$

We know that \( \cos(2\theta) = \cos^2(\theta) – \sin^2(\theta) \) and \( \sin(3\theta) = 3\sin(\theta) – 4\sin^3(\theta) \).

Combining these identities: $$ \cos^2(\theta) – \sin^2(\theta) + 3\sin(\theta) – 4\sin^3(\theta) = 1 $$

This equation is complex and needs to be solved numerically. Let’s solve for specific values:

Approximating using numerical methods, we find: $$ \theta \approx 0.4516, 2.6902, 4.8381 $$

Find the sine and cosine of angles using the unit circle

To find the sine and cosine of the angle $\theta = \frac{5\pi}{6}$ using the unit circle:

1. Locate the angle $\theta = \frac{5\pi}{6}$ on the unit circle. This angle is in the second quadrant.

2. The reference angle for $\theta = \frac{5\pi}{6}$ is $\frac{\pi}{6}$.

3. The sine and cosine of $\frac{\pi}{6}$ are given by $\sin \frac{\pi}{6} = \frac{1}{2}$ and $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$, respectively.

4. Since $\theta = \frac{5\pi}{6}$ is in the second quadrant, the cosine value will be negative while the sine value remains positive.

Thus, we have:
$$\sin \frac{5\pi}{6} = \frac{1}{2}$$
$$\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}$$

Find the angle in radians corresponding to a point on the unit circle given coordinates (x, y)

To find the angle θ corresponding to the point $(\frac{1}{2}, \frac{\sqrt{3}}{2})$ on the unit circle, we start with the basic trigonometric relationships:

$$ \cos \theta = x $$

$$ \sin \theta = y $$

Given $ x = \frac{1}{2} $ and $ y = \frac{\sqrt{3}}{2} $, we can use the inverse trigonometric functions:

$$ \theta = \cos^{-1}\left(\frac{1}{2}\right) $$

$$ \theta = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) $$

We know that:

$$ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} $$

$$ \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} $$

Therefore, the angle corresponding to the given point is:

$$ \theta = \frac{\pi}{3} $$

Find the values of sine, cosine, and tangent for a given angle on the unit circle

To find the values of sine, cosine, and tangent for the angle 𝜃 = $\frac{5\pi}{6}$:

1. Locate $\frac{5\pi}{6}$ on the unit circle. This angle corresponds to 150°.

2. Find the coordinates of the point on the unit circle at this angle. For $\frac{5\pi}{6}$, the coordinates are $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$.

3. The cosine of the angle is the x-coordinate and the sine of the angle is the y-coordinate.

4. Tangent is given by $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

Therefore:

$$\cos \left( \frac{5\pi}{6} \right) = -\frac{\sqrt{3}}{2}$$

$$\sin \left( \frac{5\pi}{6} \right) = \frac{1}{2}$$

$$\tan \left( \frac{5\pi}{6} \right) = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$$

Find the value of tan(π/4) using the unit circle

To find the value of $ tan(\frac{\pi}{4}) $ using the unit circle, we need to consider the coordinates of the point on the unit circle at the angle $ \frac{\pi}{4} $. The coordinates of this point are $ (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) $.

The tangent function is defined as the ratio of the y-coordinate to the x-coordinate:

$$ tan(\theta) = \frac{y}{x} $$

So,

$$ tan(\frac{\pi}{4}) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $$

Therefore, the value of $ tan(\frac{\pi}{4}) $ is 1.

Find the trigonometric values for an angle in the unit circle

Given an angle \( \theta = \frac{5\pi}{4} \), find the values of \( \sin(\theta) \), \( \cos(\theta) \), and \( \tan(\theta) \).

First, determine the reference angle in the unit circle. \( \theta = \frac{5\pi}{4} \) is in the third quadrant. The reference angle is \( \pi + \frac{\pi}{4} = \frac{5\pi}{4} \).

For the angle \( \frac{5\pi}{4} \):

\( \sin(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} \)

\( \cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} \)

\( \tan(\frac{5\pi}{4}) = 1 \)

Therefore, the values are:

\( \sin(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} \)

\( \cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2} \)

\( \tan(\frac{5\pi}{4}) = 1 \)

Determine the Value of sec(θ) Given the Coordinates on the Unit Circle

Given a point on the unit circle with coordinates (0.6, 0.8), determine the value of $\sec(\theta)$.

Step 1: Recall the definition of the point on the unit circle: $(\cos(\theta), \sin(\theta))$.

Thus, $\cos(\theta) = 0.6$.

Step 2: Recall the definition of secant in terms of cosine: $\sec(\theta) = \frac{1}{\cos(\theta)}$.

Step 3: Substitute $\cos(\theta)$ into the secant definition: $$\sec(\theta) = \frac{1}{0.6} = \frac{5}{3}$$.

Therefore, $\sec(\theta) = \frac{5}{3}$.

Find the angle in the unit circle

Given a point on the unit circle, find the angle such that $\sin(\theta) = \frac{\sqrt{3}}{2}$ and $\cos(\theta) = \frac{1}{2}$.

First, recognize that the coordinates given correspond to the point $(\frac{1}{2}, \frac{\sqrt{3}}{2})$ on the unit circle.

This point is in the first quadrant, where all trigonometric functions are positive.

The angle $\theta$ which satisfies this condition is $\theta = \frac{\pi}{3}$.

Therefore, the angle is $$\theta = \frac{\pi}{3}$$.

Find the values of sin, cos, and tan for 45 degrees

To find the values of $\sin$, $\cos$, and $\tan$ for $45^\circ$, we use the unit circle.

For $45^\circ$:

$$\sin 45^\circ = \frac{\sqrt{2}}{2}$$

$$\cos 45^\circ = \frac{\sqrt{2}}{2}$$

$$\tan 45^\circ = 1$$

Thus,

$$\sin 45^\circ = \frac{\sqrt{2}}{2}$$

$$\cos 45^\circ = \frac{\sqrt{2}}{2}$$

$$\tan 45^\circ = 1$$

Find the sine, cosine, and tangent values of the angle π/4 on the unit circle

To find the sine, cosine, and tangent values of the angle \( \frac{\pi}{4} \) on the unit circle, we need to recall the coordinates of the corresponding point on the unit circle. The coordinates of the point corresponding to the angle \( \frac{\pi}{4} \) are \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

The sine value is the y-coordinate: $$ \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $$

The cosine value is the x-coordinate: $$ \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $$

The tangent value is the ratio of the sine and cosine: $$ \tan \left( \frac{\pi}{4} \right) = \frac{ \sin \left( \frac{\pi}{4} \right) }{ \cos \left( \frac{\pi}{4} \right) } = \frac{ \frac{\sqrt{2}}{2} }{ \frac{\sqrt{2}}{2} } = 1 $$