Math / Trigonometry

QuestionThe Washington Monument is 555 ft tall. The angle of elevation from the end of the monument's shadow to the top of the monument has a cosecant of 1.10. a. $\theta=$ $\square$ (Type your answer in degrees. Rou

Studdy Solution

STEP 1

What is this asking? We need to find the angle of elevation, $\theta$ , at which someone would look up to see the top of the Washington Monument from the end of its shadow, knowing the monument's height and the cosecant of the angle. Watch out! Remember, the cosecant is related to the sine, not the cosine!
Don't mix them up!

STEP 2

1. Relate cosecant to sine
2. Calculate sine of theta
3. Find theta

STEP 3

Alright, let's start with what we know: the cosecant of $\theta$ is 1.10.
Remember that the cosecant of an angle is the reciprocal of its sine.
Mathematically, we can write this as: $\csc(\theta) = \frac{1}{\sin(\theta)}$

STEP 4

We're given that $\csc(\theta) = \mathbf{1.10}$ , so let's plug that in: $\mathbf{1.10} = \frac{1}{\sin(\theta)}$

STEP 5

To find the sine of $\theta$ , we need to isolate $\sin(\theta)$ .
We can do this by multiplying both sides of the equation by $\sin(\theta)$ and then dividing both sides by 1.10: $\sin(\theta) \cdot \mathbf{1.10} = 1$ $\sin(\theta) = \frac{1}{\mathbf{1.10}}$

STEP 6

Now, let's calculate the value: $\sin(\theta) \approx \mathbf{0.9091}$

STEP 7

Fantastic! We've got the sine of $\theta$ .
Now, to find $\theta$ itself, we'll use the arcsine function (also known as the inverse sine), which is written as $\sin^{-1}$ .
This function essentially "undoes" the sine function. $\theta = \sin^{-1}(\mathbf{0.9091})$

STEP 8

Using a calculator (make sure it's in degree mode!), we find: $\theta \approx \mathbf{65.38^\circ}$

STEP 9

The angle of elevation, $\theta$ , is approximately $\mathbf{65.38^\circ}$ .

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QuestionThe Washington Monument is 555 ft tall. The angle of elevation from the end of the monument's shadow to the top of the monument has a cosecant of 1.10. a. $\theta=$ $\square$ (Type your answer in degrees. Rou

Studdy Solution

STEP 1

STEP 2

1. Relate cosecant to sine
2. Calculate sine of theta
3. Find theta

STEP 3

Alright, let's start with what we know: the cosecant of $\theta$ is 1.10.
Remember that the cosecant of an angle is the reciprocal of its sine.
Mathematically, we can write this as: $\csc(\theta) = \frac{1}{\sin(\theta)}$

STEP 4

We're given that $\csc(\theta) = \mathbf{1.10}$ , so let's plug that in: $\mathbf{1.10} = \frac{1}{\sin(\theta)}$

STEP 5

To find the sine of $\theta$ , we need to isolate $\sin(\theta)$ .
We can do this by multiplying both sides of the equation by $\sin(\theta)$ and then dividing both sides by 1.10: $\sin(\theta) \cdot \mathbf{1.10} = 1$ $\sin(\theta) = \frac{1}{\mathbf{1.10}}$

STEP 6

Now, let's calculate the value: $\sin(\theta) \approx \mathbf{0.9091}$

STEP 7

Fantastic! We've got the sine of $\theta$ .
Now, to find $\theta$ itself, we'll use the arcsine function (also known as the inverse sine), which is written as $\sin^{-1}$ .
This function essentially "undoes" the sine function. $\theta = \sin^{-1}(\mathbf{0.9091})$

STEP 8

Using a calculator (make sure it's in degree mode!), we find: $\theta \approx \mathbf{65.38^\circ}$

STEP 9

The angle of elevation, $\theta$ , is approximately $\mathbf{65.38^\circ}$ .

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QuestionThe Washington Monument is 555 ft tall. The angle of elevation from the end of the monument's shadow to the top of the monument has a cosecant of 1.10. a. θ=\theta=θ= □\square□ (Type your answer in degrees. Rou

Studdy Solution

STEP 1

STEP 2

1. Relate cosecant to sine2. Calculate sine of theta3. Find theta

STEP 3

Alright, let's **start** with what we know: the cosecant of θ\thetaθ is **1.10**.Remember that the cosecant of an angle is the reciprocal of its sine.Mathematically, we can write this as: csc⁡(θ)=1sin⁡(θ) \csc(\theta) = \frac{1}{\sin(\theta)} csc(θ)=sin(θ)1​

STEP 4

We're given that csc⁡(θ)=1.10\csc(\theta) = \mathbf{1.10}csc(θ)=1.10, so let's plug that in: 1.10=1sin⁡(θ) \mathbf{1.10} = \frac{1}{\sin(\theta)} 1.10=sin(θ)1​

STEP 5

STEP 6

Now, let's **calculate** the value: sin⁡(θ)≈0.9091 \sin(\theta) \approx \mathbf{0.9091} sin(θ)≈0.9091

STEP 7

STEP 8

Using a calculator (make sure it's in degree mode!), we find: θ≈65.38∘ \theta \approx \mathbf{65.38^\circ} θ≈65.38∘

STEP 9

The angle of elevation, θ\thetaθ, is approximately 65.38∘\mathbf{65.38^\circ}65.38∘.

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QuestionThe Washington Monument is 555 ft tall. The angle of elevation from the end of the monument's shadow to the top of the monument has a cosecant of 1.10. a. $\theta=$ $\square$ (Type your answer in degrees. Rou

1. Relate cosecant to sine
2. Calculate sine of theta
3. Find theta

Alright, let's start with what we know: the cosecant of $\theta$ is 1.10.
Remember that the cosecant of an angle is the reciprocal of its sine.
Mathematically, we can write this as: $\csc(\theta) = \frac{1}{\sin(\theta)}$

We're given that $\csc(\theta) = \mathbf{1.10}$ , so let's plug that in: $\mathbf{1.10} = \frac{1}{\sin(\theta)}$

Now, let's calculate the value: $\sin(\theta) \approx \mathbf{0.9091}$

Using a calculator (make sure it's in degree mode!), we find: $\theta \approx \mathbf{65.38^\circ}$

The angle of elevation, $\theta$ , is approximately $\mathbf{65.38^\circ}$ .