Math / Trigonometry

Question3) Solve the following for both angles between $0^{\circ}$ and $360^{\circ}$ . a) $\csc \theta=2.564$ b) $\sec \theta=3.723$ c) $\cot \theta=-1.149$

Studdy Solution

STEP 1

What is this asking? We need to find the angles $\theta$ where the cosecant, secant, and cotangent equal the given values, within one full rotation. Watch out! Remember, there are usually two angles in a full rotation that satisfy these kinds of equations!
Also, make sure your calculator is in degree mode!

STEP 2

1. Solve for Cosecant
2. Solve for Secant
3. Solve for Cotangent

STEP 3

Alright, let's tackle $\csc \theta = \mathbf{2.564}$ !
Remember that cosecant is the reciprocal of sine, so we can rewrite this as $\sin \theta = \frac{1}{\mathbf{2.564}}$ .

STEP 4

Let's calculate that: $\sin \theta = \frac{1}{\mathbf{2.564}} \approx \mathbf{0.3899}$ .
Now, we're looking for an angle whose sine is approximately $\mathbf{0.3899}$ .

STEP 5

Time to grab our calculators!
Using the inverse sine function ( $\arcsin$ or $\sin^{-1}$ ), we find $\theta \approx \mathbf{22.94^\circ}$ .
This is our principal angle.

STEP 6

But hold on!
There's another angle within $0^\circ$ to $360^\circ$ where the sine is also $\mathbf{0.3899}$ .
Think about the unit circle: sine is positive in the first and second quadrants.
The second angle is $180^\circ - \mathbf{22.94^\circ} = \mathbf{157.06^\circ}$ .

STEP 7

Next up: $\sec \theta = \mathbf{3.723}$ .
Secant is the reciprocal of cosine, so $\cos \theta = \frac{1}{\mathbf{3.723}}$ .

STEP 8

Calculating that gives us $\cos \theta = \frac{1}{\mathbf{3.723}} \approx \mathbf{0.2686}$ .
We're hunting for the angle whose cosine is approximately $\mathbf{0.2686}$ .

STEP 9

Using the inverse cosine function ( $\arccos$ or $\cos^{-1}$ ), we get $\theta \approx \mathbf{74.34^\circ}$ .
This is our principal angle.

STEP 10

Cosine is positive in the first and fourth quadrants.
So, our second angle is $360^\circ - \mathbf{74.34^\circ} = \mathbf{285.66^\circ}$ .

STEP 11

Finally, we have $\cot \theta = \mathbf{-1.149}$ .
Cotangent is the reciprocal of tangent, meaning $\tan \theta = \frac{1}{\mathbf{-1.149}}$ .

STEP 12

Calculating that gives $\tan \theta = \frac{1}{\mathbf{-1.149}} \approx \mathbf{-0.8703}$ .
We're looking for the angle whose tangent is approximately $\mathbf{-0.8703}$ .

STEP 13

Using the inverse tangent function ( $\arctan$ or $\tan^{-1}$ ), we find $\theta \approx \mathbf{-41.03^\circ}$ .
Since we want angles between $0^\circ$ and $360^\circ$ , we add $360^\circ$ to get $\theta = \mathbf{-41.03^\circ} + 360^\circ = \mathbf{318.97^\circ}$ .

STEP 14

Tangent is negative in the second and fourth quadrants.
We already found an angle in the fourth quadrant.
For the second quadrant, we can add $180^\circ$ to our original negative angle: $\mathbf{-41.03^\circ} + 180^\circ = \mathbf{138.97^\circ}$ .

STEP 15

a) $\theta \approx 22.94^\circ$ and $\theta \approx 157.06^\circ$ b) $\theta \approx 74.34^\circ$ and $\theta \approx 285.66^\circ$ c) $\theta \approx 138.97^\circ$ and $\theta \approx 318.97^\circ$

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