What does the unit circle tell us?

The triangle at the right is a right triangle, and θ is in standard position.

Fill in the Blank

What is the length of the hypotenuse of the triangle? (Think about it as a radius)

OP=___

Label OP on your paper.

Match

Using θ as a reference angle, mark the triangle sides as Opp, Adj, and Hyp.

Label these on the triangle on your paper.

• x

• y

• OP

Adj

Opp

Hyp

• Adj

• Opp

• Hyp

Match

Now, use the given triangle to represent $$\sin\theta$$ and $$\cos\theta$$ in terms of the variables given in the diagram.

• $$\frac{y}{1}=y$$

• $$\frac{x}{1}=x$$

• 1

sinθ

cosθ

OP

• sinθ

• cosθ

• OP

Drag and Drop

What about tangent? Use the triangle to answer the following questions.



In terms of x and y, what does tanθ equal?

tanθ=​ ​

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In terms of sinθ and cosθ, what does tanθ equal?

tanθ=​

Placeholder

Drag these tiles and drop them in the correct blank above

x

y

sinθ

cosθ

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Look at YOUR unit circle. What is the reference angle for $$\frac{4\pi}{3}$$ ?



Reference angle = ​

Select answer

Dropdown

Look at YOUR unit circle. What is the reference angle for $$\frac{7\pi}{4}$$ ?



Reference angle = ​ ​

Select answer

Drag and Drop

Unlike right triangle trigonometry, sine and cosine may be negative.

Thinking about how sine, cosine, and tangent relate to (x,y) coordinate pairs on the unit circle, find the quadrants where:



sine is positive: ​

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​&

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sine is negative: ​ ​

Placeholder

&​

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Drag these tiles and drop them in the correct blank above

I

II

III

IV

Drag and Drop

Unlike right triangle trigonometry, sine and cosine may be negative.

Thinking about how sine, cosine, and tangent relate to (x,y) coordinate pairs on the unit circle, find the quadrants where:



cosine is positive: ​

Placeholder

​&

Placeholder

cosine is negative: ​ ​

Placeholder

&​

Placeholder

Drag these tiles and drop them in the correct blank above

I

IV

II

III

Drag and Drop

Unlike right triangle trigonometry, sine and cosine may be negative.

Thinking about how sine, cosine, and tangent relate to (x,y) coordinate pairs on the unit circle, find the quadrants where:



tangent is positive: ​

Placeholder

​&

Placeholder

tangent is negative: ​ ​

Placeholder

&​

Placeholder

​

Drag these tiles and drop them in the correct blank above

I

III

II

IV

Dropdown

Find the exact value of the following trig expressions.



Try to remember these results without looking at the unit circle.



$$\sin\left(\frac{\pi}{3}\right)=$$ ​

Select answer

$$\sin\left(\frac{7\pi}{6}\right)=$$ ​ ​ ​

Select answer

$$\cos\left(\frac{5\pi}{4}\right)=$$ ​

Select answer

$$\cos\left(-\frac{\pi}{3}\right)=$$ ​

Select answer

The unit circle is a great tool for finding the exact value of trig functions, but it is not your only tool. Sometimes it is still convenient to think of using right triangles.

Fill in the Blank

Suppose you know that for a particular angle, $$\cos\theta=\frac{4}{5}$$ . The related triangle is shown.

Find the missing side length.

Dropdown

Suppose you know that for a particular angle, $$\cos\theta=\frac{4}{5}$$ . The related triangle is shown.



Given that the missing side has length 3,

$$\sin\theta=$$ ​

Select answer

Fill in the Blank

Suppose you know that for a particular angle, $$\cos\theta=\frac{4}{5}$$ . The related triangle is shown.

If this triangle were superimposed on a circle so that the vertex of $$\theta$$ was the center of the circle, what would be the radius of the circle?

Multiple Choice

Suppose you know that for a particular angle, $$\cos\theta=\frac{4}{5}$$ . The related triangle is shown.

If we know that $$\frac{3\pi}{2}<\theta<2\pi$$ (i.e., the angle lies between $$\frac{3\pi}{2}$$ radians and $$2\pi$$ radians), what quadrant on the unit circle is the angle $$\theta$$ in?

• I

• II

• III

• IV

Fill in the Blank

Suppose you know that for a particular angle, $$\cos\theta=\frac{4}{5}$$ . The related triangle is shown.

Given that $$\theta$$ is in the fourth quadrant,

$$\sin\theta=$$ ____

(type your answer as a fraction)