Special Right Triangles and the Unit Circle

Special right triangles

A 45-45-90 triangle has two equal legs. If each leg has length $a$ , the hypotenuse is $a\sqrt{2}$ .

A 30-60-90 triangle has a short leg opposite the 30 degree angle. If the short leg has length $a$ , the long leg is $a\sqrt{3}$ and the hypotenuse is $2a$ .

To find a missing side, write the side you know as a multiple of $a$ , solve for $a$ , and then use the ratio for the side you need. Leave the answer in exact form, and rationalize the denominator when a square root is left in it.

8 questions.

The unit circle

The unit circle is the circle of radius 1 centered at the origin. The point where an angle $\theta$ meets the circle is $(\cos\theta, \sin\theta)$ , so cosine is the x-coordinate and sine is the y-coordinate. Tangent is $\tan\theta = \frac{\sin\theta}{\cos\theta}$ , which is undefined when $\cos\theta = 0$ .

The values for the angles in the first quadrant, and for $\pi$ , are below. Every other special angle uses one of these values with a sign that depends on the quadrant.

$θ$	$0$	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$	$π$
$sin θ$	$\frac{0}{2}$	$\frac{1}{2}$	$\frac{2}{2}$	$\frac{3}{2}$	$\frac{4}{2}$	$\frac{0}{2}$
$cos θ$	$\frac{4}{2}$	$\frac{3}{2}$	$\frac{2}{2}$	$\frac{1}{2}$	$\frac{0}{2}$	$- \frac{4}{2}$
$tan θ$	$0$	$\frac{3}{3}$	$1$	$3$	$undefined$	$0$

The numerator pattern rule

For standard non-axis angles with denominator 3, 4, or 6, the numerator $N$ follows a pattern based on the quadrant. Write $D$ for the denominator.

Quadrant I: $N = 1$ . The fraction is less than $\frac{1}{2}$ . Examples: $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$ .
Quadrant II: $N = D - 1$ . The numerator is 1 less than the denominator, just before $\pi$ . Examples: $\frac{5\pi}{6}, \frac{3\pi}{4}, \frac{2\pi}{3}$ .
Quadrant III: $N = D + 1$ . The numerator is 1 more than the denominator, just after $\pi$ . Examples: $\frac{7\pi}{6}, \frac{5\pi}{4}, \frac{4\pi}{3}$ .
Quadrant IV: $N = 2D - 1$ . The numerator is 1 less than double the denominator, just before $2\pi$ . Examples: $\frac{11\pi}{6}, \frac{7\pi}{4}, \frac{5\pi}{3}$ .

Angles are always given in radians here, never degrees. Practice finding a point from its radian angle, then giving its coordinates.

Click the point at the given angle, then type its coordinates. 10 rounds.

10 questions.

Signs by quadrant

Sine is positive in quadrants I and II, cosine is positive in quadrants I and IV, and tangent is positive in quadrants I and III. Use the quadrant to choose the sign, and use the reference angle to choose the value.

8 questions.

Negative and coterminal angles

A positive angle turns counterclockwise from the positive x-axis, and a negative angle turns clockwise. Two angles are coterminal when they differ by a whole number of full turns, so adding or subtracting $2\pi$ gives a coterminal angle with the same trigonometric values.

For a negative angle, $\sin(-\theta) = -\sin\theta$ , $\cos(-\theta) = \cos\theta$ , and $\tan(-\theta) = -\tan\theta$ .

8 questions.

Inverse trigonometric functions

An inverse trigonometric function returns an angle in a fixed range.

$\arcsin x$ returns an angle from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ .
$\arccos x$ returns an angle from $0$ to $\pi$ .
$\arctan x$ returns an angle between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ , not including the endpoints.

To evaluate one, find the angle in that range whose trigonometric value equals the input.

8 questions.

Solving trigonometric equations on an interval

To find all $x$ in an interval such as $0 \le x \le \pi$ that satisfy an equation, find the reference angle for the value, use the sign to find which quadrants apply, and list every angle in the interval. Give the answers separated by commas.

8 questions.

Review

This challenge mixes every question type from this lesson.

18 questions.