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# Angles of Rotation in Standard Positions

## Counterclockwise angles beginning at the positive x-axis.

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Angles, Bearings, and the Angles in the Unit Circle

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Angles of Elveation and Depression

Your trig functions will come in handy when solving problems that involve angles of elevation or depression

Remember: SOHCAHTOA

[Figure1]

Angle of depression and elevation are equal to eachother because they are alternate interior angles.

Angle of Depression: is the angle formed by a horizontal line and the line of sight down to an object when the image of an object is located beneath the horizontal line.

Angle of Elevation:  is the angle formed by a horizontal line and the line of sight up to an object when the image of an object is located above the horizontal line.

[Figure2]

If you are given and angle and a side, what tools can you use to solve for the other sides?

Angle of elevation and depression will normally involve a right triangle!

A pilot looks down at an angle of depression of 30o and sees a 6-foot man standing on flat ground 9135 feet away away. How high is the airplane in miles? (Answer Below)

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We want to find the height of the airplane, h.
First, we are given the angle of depression which we know is equal to the angle of elevation.
We also know that because the man is 6-feet tall that Y= h-6.

Recall that tangent is opposite over adjacent. Therefore, because we are given an angle, it's adjacent, and we are trying to solve for the angle's opposite, we can use this ratio to set up the equations below.

tan(30)=Y9135feet\begin{align*}tan(30^\circ )=\frac{Y}{9135 feet} \end{align*}

13=Y9135feet\begin{align*} \frac{1}{\sqrt{3}}=\frac{Y}{9135 feet} \end{align*}

9135feet3=Y\begin{align*} \frac{9135 feet}{\sqrt{3}}=Y\end{align*}

Y5274feet\begin{align*}Y\approx5274 feet\end{align*}

h=Y+6\begin{align*}h=Y+6\end{align*}

h=5280feet\begin{align*}h=5280 feet\end{align*}
h=5280feet×1mile5280feet\begin{align*}h= 5280 feet\times\frac{1mile}{5280 feet}\end{align*}
h=1mile\begin{align*}h=1 mile\end{align*}

Bearings

Bearing: is the direction taken to get from one place to another.

Always sketch a coordinate axis before you begin to start the problem, it will make the drawing easier

How are the bearing problems similar to the angles of elevation problems?

[Figure4]

*Think of the coordinate axis as a compass*

Pay close attention to the direction that the angle is being place, it will normally start from north or south and then either west or east.

Use SOHCAHTOA to find the missing side since the problem is most likely going to involve a right triangle.

Angles of rotation

A standard positioned angle means it is always positive because it always goes counter-clockwise in the graphing table.

A quadrantal angle  is an angle whose terminal side lies along either the positive or negative 'x' axis or the positive or negative 'y' axis

Coterminal Angles

Coterminal Angles: are angles with the same terminal side but expressed differently, such as a different number of complete rotations around the unit circle or angles being expressed as positive versus negative angle measurements.

There are a infinitely number of coterminal sides for one angle, what does this mean when it comes to answering the problem?

What trigonometric functions can you define using the unit circle?

Angles in the Unit Circle

By using the Unit Circle, you are able to apply Trig Function to angles that are greater than 90o , because of the special angles!

How are the trig functions used when it comes to the Unit Circle?

What were the angles that make up special triangles?

[Figure5]

These are the special angles you can use to be able to apply trig funtions for angles that are 90o because you can connect the angles using their coterminal angles.

Reference Angle: is the angle formed between the terminal side of an angle and the closest of either the positive or negative 'x' axis.

What is the difference/similarity of a reference angle and a coternminal angle?

A negative angle will can always have a  coterminal angle or refererece angle which can be used to aplly Trig Functions.

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