Back to Exercise: Use special triangles and unit circle

Exercises: Exact Trigonometric Values from Special Triangles

Use special triangles and unit circle symmetry. Express all answers in exact form (no decimals).

Grade 9·22 problems·~30 min·Common Core Math - HS Functions·group·hsf-tf-a-3

Work through problems with immediate feedback

Warm-Up: Review What You Know

These problems review skills you have already learned.

A right triangle has legs of length 3 and 4. What is the length of its hypotenuse?

$\sqrt{7}$

$\sqrt{25}$

$Unit circle with a first-quadrant point P at (a, b), showing horizontal distance a and vertical distance b$

A point $P$ lies on the unit circle at angle $\theta$ . Its coordinates are $(a, b)$ . Which statements are correct?

$\cos\theta = a$ and $\sin\theta = b$

$\cos\theta = b$ and $\sin\theta = a$

$\cos\theta = \frac{a}{b}$ and $\sin\theta = \frac{b}{a}$

$\cos\theta = \frac{1}{a}$ and $\sin\theta = \frac{1}{b}$

The terminal side of angle $\frac{5\pi}{4}$ lies in the third quadrant. What is its reference angle? Express your answer in the form $\frac{a\pi}{b}$ (e.g., pi/4).

Fluency Practice

Find the exact value. Show the special triangle you used.

$45-45-90 triangle inscribed in the unit circle with legs sqrt(2)/2 and hypotenuse 1$

A 45-45-90 triangle is placed in the unit circle with its hypotenuse along the radius to angle $\frac{\pi}{4}$ . Each leg has length $\frac{\sqrt{2}}{2}$ . What is $\cos\!\left(\frac{\pi}{4}\right)$ ?

$30-60-90 triangle inscribed in the unit circle with the 30-degree angle at origin, vertical leg 1/2, horizontal leg sqrt(3)/2$

An equilateral triangle with side length 2 is bisected to form a 30-60-90 triangle with sides 1, $\sqrt{3}$ , and 2. When scaled to hypotenuse 1 and placed in the unit circle at angle $\frac{\pi}{6}$ , what is $\sin\!\left(\frac{\pi}{6}\right)$ ?

Using the 30-60-90 triangle on the unit circle, find the exact value of $\cos\!\left(\frac{\pi}{3}\right)$ .

Find the exact value of $\tan\!\left(\frac{\pi}{6}\right)$ . Express your answer as a fraction with a rational denominator.

Complete the unit circle coordinates for the three first-quadrant special angles:

Angle $\frac{\pi}{6}$ : point $= \left(\_\_\_, \, \_\_\_\right)$
Angle $\frac{\pi}{4}$ : point $= \left(\_\_\_, \, \_\_\_\right)$
Angle $\frac{\pi}{3}$ : point $= \left(\_\_\_, \, \_\_\_\right)$

cos(pi/6):

sin(pi/6):

cos(pi/4):

sin(pi/4):

cos(pi/3):

sin(pi/3):

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