Trigonometry

Unit-circle values

ALEKS placement

Use special triangles or the axes for the sizes, then use the quadrant for the signs.

What this covers

sin30∘=21,sin45∘=2√2,sin60∘=2√3

Find the reference angle: the acute angle to the nearest x-axis.
For 30°, 45°, and 60°, the sine values are 1/2, √2/2, and √3/2 in that order.
Cosine uses that same list backwards. Tangent is sine divided by cosine.
In radians those three are π/6, π/4, π/3. On the axes themselves, sine and cosine are only ever 0, 1, or -1.
Attach the sign from the quadrant — All, Sine, Tangent, Cosine are positive in order.
A point on the circle is (cos θ, sin θ).

Worked example

Find cos(5π/6).

reference=6π5π/6 is π/6 short of π.
cos6π=2√3That is the 30-60-90 value.
−2√35π/6 sits in quadrant II, where cosine is negative.

Another worked example

Find tan(5π/4).

(cosθ,sinθ)=(−2√2,−2√2)The angle is 225°, in quadrant III, with reference angle 45°. Both coordinates are negative.
tan45π=−√2/2−√2/2=1Tangent is sine divided by the nonzero cosine; the matching negative values divide to positive 1.

A common mistake

A mistake Lemma catches

From cos6π=2√3

cos65π=2√3cos65π=−2√3

“π/6 is only the reference angle; it sets the size, √3/2. 5π/6 lies in quadrant II, where x-coordinates — cosines — are negative, so cos(5π/6) = −√3/2.”

Try one

Sample problem

Evaluate exactly.

cos(35π)

Practice unit circle free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

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