5.1 Angles, radians, and right triangles

Angles in standard position begin on the positive \(x\)-axis. Positive rotation is counterclockwise and negative rotation is clockwise. Coterminal angles differ by whole revolutions.

5.1.1 Degree and radian measure

Radian. The central angle that intercepts an arc equal in length to the circle’s radius.

One revolution is \(360^\circ=2\pi\) radians. Convert degrees to radians by multiplying by \(\pi/180^\circ\), and radians to degrees by multiplying by \(180^\circ/\pi\).

Worked Example: Converting an angle

\[225^\circ\left(\frac{\pi}{180^\circ}\right)=\frac{5\pi}{4}.\]

Try It. Convert \(150^\circ\) to radians.
Check Your Work \(5\pi/6\).

Worked Example: Finding a coterminal angle

\(-116^\circ+360^\circ=244^\circ\), so \(244^\circ\) is a positive coterminal angle.

Try It. Find a positive coterminal angle for \(-5\pi/6\).
Check Your Work \(-5\pi/6+2\pi=7\pi/6\).

5.1.2 Right-triangle ratios

For an acute angle \(\theta\), \(\sin\theta=\) opposite/hypotenuse, \(\cos\theta=\) adjacent/hypotenuse, and \(\tan\theta=\) opposite/adjacent. Their reciprocals are cosecant, secant, and cotangent.

Worked Example: Recovering all six ratios

If \(\sin\theta=2/5\), take opposite side 2 and hypotenuse 5. The adjacent side is \(\sqrt{21}\). Thus \(\cos\theta=\sqrt{21}/5\), \(\tan\theta=2/\sqrt{21}\), and the reciprocal ratios follow.

Try It. If \(\cos\theta=3/5\), find \(\sin\theta\) and \(\tan\theta\) for acute \(\theta\).
Check Your Work The triangle is 3-4-5, so \(\sin\theta=4/5\) and \(\tan\theta=4/3\).

Worked Example: Finding a pipe rise

A 14 m pipe is inclined at \(8^\circ\). Its vertical rise is

\[14\sin8^\circ\approx1.95\ \text{m}.\]

Try It. Find the rise of a 20 m pipe inclined at \(5^\circ\).
Check Your Work \(20\sin5^\circ\approx1.74\) m.
Applied Problem: Surveying a tank. From a point 32 m from a tank, the angle of elevation to its top is \(28^\circ\). The instrument is 1.6 m above ground. Find the tank height.
Check Your Work The rise above the instrument is \(32\tan28^\circ\approx17.0\) m. Adding 1.6 m gives approximately 18.6 m.

5.1.3 Practice Problems

  1. Convert \(300^\circ\) to radians.
    Check Your Work \(5\pi/3\).
  2. Convert \(7\pi/12\) to degrees.
    Check Your Work \(105^\circ\).
  3. Find a positive coterminal angle for \(-70^\circ\).
    Check Your Work \(290^\circ\).
  4. In a right triangle, opposite \(=6\) and adjacent \(=8\). Find \(\tan\theta\).
    Check Your Work \(3/4\).
  5. Find the hypotenuse when the legs are 5 and 12.
    Check Your Work 13.
  6. A channel bank rises 2.4 m over a horizontal run of 7.0 m. Find its angle.
    Check Your Work \(\theta=\arctan(2.4/7.0)\approx18.9^\circ\).
  7. A 9 m ladder reaches a tank platform at \(65^\circ\). Find the vertical reach.
    Check Your Work \(9\sin65^\circ\approx8.16\) m.
  8. A pipe falls 0.8 m over 24 m horizontally. Find the downward angle.
    Check Your Work \(\arctan(0.8/24)\approx1.91^\circ\).