5.2 Special angles, reference angles, and the unit circle

Special triangles and the unit circle provide exact trigonometric values without calculator approximations.

5.2.1 Special triangles and reference angles

The \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle has side ratio \(1:1:\sqrt2\). The \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle has ratio \(1:\sqrt3:2\). A reference angle is the acute angle between a terminal side and the \(x\)-axis.

Worked Example: Evaluating a special angle

From the \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle, \(\sin60^\circ=\sqrt3/2\), \(\cos60^\circ=1/2\), and \(\tan60^\circ=\sqrt3\).

Try It. Give the exact values of sine and cosine at \(45^\circ\).
Check Your Work Both equal \(\sqrt2/2\).

Worked Example: Finding a reference angle

The angle \(240^\circ\) lies in quadrant III, so its reference angle is \(240^\circ-180^\circ=60^\circ\).

Try It. Find the reference angle for \(11\pi/6\).
Check Your Work \(2\pi-11\pi/6=\pi/6\).

5.2.2 Trigonometric functions of any angle

For a point \((x,y)\) on a terminal side with \(r=\sqrt{x^2+y^2}\), \(\sin\theta=y/r\), \(\cos\theta=x/r\), and \(\tan\theta=y/x\). On the unit circle, the point is \((\cos\theta,\sin\theta)\). Signs follow the quadrant.

Worked Example: Evaluating with a reference angle

\(315^\circ\) has reference angle \(45^\circ\) in quadrant IV. Therefore, \(\cot315^\circ=-1\).

Try It. Evaluate \(\tan(11\pi/6)\) exactly.
Check Your Work \(-1/\sqrt3=-\sqrt3/3\).

Worked Example: Using a quadrant condition

If \(\sin\theta=-1/3\) and \(\cos\theta<0\), then \(\theta\) is in quadrant III. Using \(\cos^2\theta=1-1/9=8/9\) gives \(\cos\theta=-2\sqrt2/3\) and \(\tan\theta=\sqrt2/4\).

Try It. If \(\cos\theta=-3/5\) in quadrant II, find sine and tangent.
Check Your Work \(\sin\theta=4/5\) and \(\tan\theta=-4/3\).
Applied Problem: Resolving a force direction. A 250 N force acts at \(135^\circ\) from the positive horizontal direction. Find its horizontal and vertical components.
Check Your Work \(F_x=250\cos135^\circ=-125\sqrt2\approx-176.8\) N and \(F_y=250\sin135^\circ=125\sqrt2\approx176.8\) N.

5.2.3 Practice Problems

  1. Evaluate \(\sin30^\circ\).
    Check Your Work \(1/2\).
  2. Evaluate \(\cos(3\pi/4)\).
    Check Your Work \(-\sqrt2/2\).
  3. Find the reference angle for \(7\pi/6\).
    Check Your Work \(\pi/6\).
  4. Evaluate \(\sec240^\circ\).
    Check Your Work \(-2\).
  5. If \((x,y)=(-5,12)\), find \(\sin\theta\).
    Check Your Work \(12/13\).
  6. A pipe thrust is 80 N at \(30^\circ\). Find its horizontal component.
    Check Your Work \(80\cos30^\circ=40\sqrt3\approx69.3\) N.
  7. A sloped channel side has direction \(210^\circ\). State the signs of its horizontal and vertical components.
    Check Your Work Both are negative.
  8. A survey ray lies in quadrant IV with cosine \(5/13\). Find its sine.
    Check Your Work \(-12/13\).