Trig values at special angles
Trig Values at Special Angles
Trig Values at Special Angles is a topic in Trigonometric Functions in the Common Core State Standards. It is aligned to Standard F-TF.3, which requires students to use special triangles to determine the exact values of trigonometric functions.
The special angles \(30^\circ,45^\circ,60^\circ\) have exact trigonometric values from the 30-60-90 and 45-45-90 triangles, extended to any quadrant with reference angles.
Theory
The special angles \(30^\circ,45^\circ,60^\circ\) (\(\dfrac{\pi}{6},\dfrac{\pi}{4},\dfrac{\pi}{3}\)) have exact trig values that come from two triangles:
- 30-60-90: sides in ratio \(1:\sqrt3:2\).
- 45-45-90: sides in ratio \(1:1:\sqrt2\).
On the unit circle, these give the coordinates \((\cos\theta,\sin\theta)\) at each special angle. For angles outside the first quadrant, use the reference angle for the magnitude and the quadrant for the sign.
The exact values worth memorizing:
How to evaluate at a special angle
- Reduce to a coterminal angle in \([0,2\pi)\) if needed.
- Find the reference angle (\(30^\circ,45^\circ,\) or \(60^\circ\)).
- Look up the exact value at that reference angle.
- Attach the sign for the quadrant using ASTC.
From the 30-60-90 triangle (sides \(1:\sqrt3:2\)), the side opposite \(30^\circ\) over the hypotenuse is
| \(\sin 30^\circ\) | \(=\) | \(\dfrac{1}{2}\) |
In the 45-45-90 triangle (sides \(1:1:\sqrt2\)),
| \(\cos 45^\circ\) | \(=\) | \(\dfrac{1}{\sqrt2}=\dfrac{\sqrt2}{2}\) |
Tangent is opposite over adjacent in the 30-60-90 triangle.
| \(\tan 60^\circ\) | \(=\) | \(\dfrac{\sqrt3}{1}=\sqrt3\) |
\(\dfrac{5\pi}{6}\) is in Quadrant II (reference angle \(\dfrac{\pi}{6}\)); sine is positive there.
| \(\sin\dfrac{5\pi}{6}\) | \(=\) | \(+\sin\dfrac{\pi}{6}\) |
| \(=\) | \(\dfrac{1}{2}\) |
Common pitfalls
Frequently asked questions
What are the special angles?
\(30^\circ,45^\circ,60^\circ\) (and \(0^\circ,90^\circ\)), whose trig values are exact and come from the 30-60-90 and 45-45-90 triangles.
What is sin 30 degrees?
\(\dfrac{1}{2}\). From the 30-60-90 triangle, the side opposite \(30^\circ\) is half the hypotenuse.
How do you find trig values outside the first quadrant?
Use the reference angle for the numerical value, then attach the sign that the quadrant requires (ASTC).
What does ASTC mean?
A memory aid for signs: All, Sine, Tangent, Cosine are positive in quadrants I, II, III, IV respectively.