[Video] Triangles to Trig Functions

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This series of videos will be aimed at using basic right angle trigonometry and the Pythagorean Theorem to develop an understanding of trigonometric functions and the unit circle and understand how it can be used and how what it means, as well some typical values on the unit circle and of trigonometric functions.

The following videos will cover the unit circle:

Introduction and Question 1: The Pythagorean Theorem applications

(Click the arrow to the right to see the Introduction video and first question.)

(Click the arrow to the right to see the question posed at the end of the video.)

If the hypotenuse of a right triangle is $8$ cm long and one side is $6$ cm long, what is the length of the remaining side?

$1$ $\sqrt {2}$ $2$ $\sqrt {14}$ $2\sqrt {7}$ $6$ $8$ $14$ $28$

(Click the arrow to the right to see a hint.)

The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the remaining sides.

(Click the arrow to the right to see an example.)

Example 1

Question 2: Defining trigonometric functions via right triangles

(Click the arrow to the right to see the definition of trigonometric functions and second question.)

(Click the arrow to the right to see the question posed at the end of the video.)

Consider the triangle below where $a = 6$ , $b=2\sqrt {7}$ , and $c=8$ . What is $\sin ( \theta )$ ?

$\sin (\theta )\text { is }\answer [given]{6/8}$

(Click the arrow to the right to see a short refresher.)

Refresher 2

(Click the arrow to the right to see an example.)

Example 2

Question 3: What is a radian and converting between radians and degrees

(Click the arrow to the right to see the video about radians, converting between radians and degrees, and the third question.)

(Click the arrow to the right to see the answers to the question posed at the end of the video.)

What is $35^\circ$ in radians? $35^\circ \text { in radians is }\answer [given]{35 \pi / 180}$

(Click the arrow to the right to see a short refresher.)

Refresher 3

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Example 3

Question 4: Evaluating trigonometric functions without a triangle

(Click the arrow to the right to see the video about evaluating trigonometric functions without a triangle, and the fourth question.)

(Click the arrow to the right to see the question posed at the end of the video.)

What is $\tan \Big (\frac {\pi }{4}\Big )$ ? $\tan \Big (\frac {\pi }{4}\Big )\text { is }\answer [given]{1}$

(Click the arrow to the right to see an example.)

Example 4

Question 5: Building the unit circle in the first quadrant and understanding it

(Click the arrow to the right to see the video about how to build the unit circle, what points on the unit circle mean, and the fifth question.)

(Click the arrow to the right to see the question posed at the end of the video.)

In the first quadrant, how many solutions are there to $\cos (\theta ) = \frac {\pi }{2}$ $\text { There are }\answer [given]{0}\text { solution(s) to }\cos (\theta )=\frac {\pi }{2}\text { in the first quadrant.}$

(Click the arrow to the right to see a short refresher.)

Refresher 5

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Example 5

Question 6: Extending the unit circle and trigonometric functions to all angles

(Click the arrow to the right to see the video about extending the unit circle trigonometric functions to all angles, and the sixth question.)

(Click the arrow to the right to see the question posed at the end of the video.)

What is $\csc \Big (\dfrac {4\pi }{3}\Big )$ ? $\csc \Big ( \dfrac {4\pi }{3}\Big ) \text { is equal to } \answer [given]{-\frac {2}{\sqrt {3}}}.$

(Click the arrow to the right to see a short refresher.)

Refresher 6

(Click the arrow to the right to see an example.)

Example 6

Question 7: The First Pythagorean Identity and the Unit Circle

(Click the arrow to the right to see the video about the first Pythagorean Identity, and the seventh question.)

(Click the arrow to the right to see the question posed at the end of the video.)

What is the resulting identity if both sides of the Pythagorean Identity are multiplied by $\csc ^2(\theta )$ ? (Please simplify any products or division of trigonometric functions to one trigonometric function if possible) $1 + \answer [given]{\cot ^2(\theta )} \hspace {1cm} = \answer [given]{\csc ^2(\theta )}.$

(Click the arrow to the right to see an example.)

Example 7

Question 8: Domains of Pythagorean Identities

(Click the arrow to the right to see the video about domains of Pythagorean Identities, and the eighth question.)

(Click the arrow to the right to see the question posed at the end of the video.)

For what values of $\theta$ is it true that $1 + \cot ^2(\theta ) = \csc ^2(\theta )$ ?

For all values of $\theta$ . For all values of $\theta$ , except $0$ . For all values of $\theta$ , except $\frac {\pi }{2}$ . For all values of $\theta$ , except $\pi n$ when $n$ is an integer. For all values of $\theta$ , except $2\pi n$ when $n$ is an integer. For all values of $\theta$ , except $\frac {\pi }{2} + \pi n$ when $n$ is an integer. For all values of $\theta$ , except $\frac {\pi }{2} + 2\pi n$ when $n$ is an integer.

(Click the arrow to the right to see an example.)

Example 8