Radians
The radian measure of an angle is given by
We measure angles in radians when we work with arclength.
The radian measure of an angle is given by
Radian measure can be expressed as multiples of or as decimals.
Degrees | \(\dfrac{\text{Radians:}}{\text{Exact Values}}\) | \(\dfrac{\text{Radians: Decimal}}{\text{Approximations}}\) |
\(0\degree\) | \(0\) | \(0\) |
\(90\degree\) | \(\dfrac{\pi}{2}\) | \(1.57\) |
\(180\degree\) | \(\pi\) | \(3.14\) |
\(270\degree\) | \(\dfrac{3\pi}{2}\) | \(4.71\) |
\(360\degree\) | \(2\pi\) | \(6.28\) |
We multiply by the appropriate conversion factor to convert between degrees and radians.
To convert from radians to degrees we multiply the radian measure by \(\dfrac{180\degree}{\pi}\text{.}\)
To convert from degrees to radians we multiply the degree measure by \(\dfrac{\pi}{180}\text{.}\)
On a circle of radius \(r\text{,}\) the length \(s\) of an arc spanned by an angle \(\theta\) in radians is
You should memorize the trig values of the special angles in radians.
Degrees | Radians | Sine | Cosine | Tangent |
\(0\degree\) | \(0\) | \(0\) | \(1\) | \(0\) |
\(30\degree\) | \(\dfrac{\pi}{6}\) | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{\sqrt{3}}\) |
\(45\degree\) | \(\dfrac{\pi}{4}\) | \(\dfrac{1}{\sqrt{2}}\) | \(\dfrac{1}{\sqrt{2}}\) | \(1\) |
\(60\degree\) | \(\dfrac{\pi}{3}\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) | \(\sqrt{3}\) |
\(90\degree\) | \(\dfrac{\pi}{2}\) | \(1\) | \(0\) | undefined |
The coordinates of the point \(P\) determined by an arc of length \(t\) in standard position on a unit circle are
Let \(P\) be the terminal point of an arc of length \(t\) in standard position on a unit circle. The circular functions of \(t\) are defined by
\(f(x) = \sin x\)
\(g(x) = \cos x\)
\(h(x) = \tan x\)
For Problems 1–2, convert from degrees to radians. Give exact answers.
For Problems 3–4, convert from degrees to radians. Round to two decimal places.
For Problems 5–8, convert from radians to degrees. Round to hundredths if necessary.
For Problems 9–10, express each fraction of one revolution as an angle in radians.
For Problems 11–12, express each angle in radians as a fraction of one revolution.
For Problems 13–14, in which quadrant on a unit circle does an arc with given length lie?
A lawn sprinkler has a range of 15 feet, and waters a porion of a circle whose curved edge is 39.27 feet long. Through what angle does the sprinkler turn?
Many telecommunications satellites are put into geostationary orbits, so that they have the same period as the rotation of the earth, and hence stay in the same relative position seen from earth. Hundreds of these satellites orbit 22,300 miles above the equator in what is called the Clarke belt, named after Arthur C. Clarke. What is the speed of the satellites? (The radius of the earth is about 4000 miles.)
The planet Neptune is 4504 million kilometers from the Sun. In one Earth year (365 days), it travels a distance of 171.58 million kilometers around its orbit.
For Problems 19–20, evaluate exactly.
For Problems 21–22, factor the expressionsketch an arc with the given length in standard position on a unit circle. Find the coordinates of the terminal point. Round to tenths.
In Problems 23–24, the circle has radius \(r\text{,}\) and \(O\) is the point \((r,0)\text{.}\)
Find the coordinates of each point in terms of \(\alpha\text{.}\)
Find the length of each arc in terms of \(\alpha\text{.}\)
For Problems 25–26, find an exact value for the area of the sector.
With a central angle of \(135\degree\) in a circle of radius 4 inches.
With a central angle of \(240\degree\) in a circle of radius 12 centimeters.
If \(\dfrac{\pi}{2} \lt \alpha \lt \beta \lt \pi\text{,}\) then \(\cos \alpha \underline{\hspace{2.727272727272727em}} \cos \beta.\)
If \(\pi \lt \theta \lt \phi \lt \dfrac{3\pi}{2}\text{,}\) then \(\sin \theta \underline{\hspace{2.727272727272727em}} \sin \phi.\)
If \(\dfrac{3\pi}{2} \lt s \lt t \lt 2\pi\text{,}\) then \(\tan s \underline{\hspace{2.727272727272727em}} \tan t.\)
If \(\dfrac{\pi}{2} \lt x \lt y \lt \dfrac{3\pi}{2}\text{,}\) then \(\cos x \underline{\hspace{2.727272727272727em}} \cos y.\)
For Problems 31–34, evaluate the function
\(f(t) = 12 - 2.8\sin (3.5t - 2)\) for \(t = 8\)
\(h(x) = 2.4 + 6\tan(\dfrac{3x-5}{4})\) for \(x = 1.8\)
\(g(z) = 0.07\tan(0.4z + 0.2) - 1.3\) for \(z = 22\)
\(F(s) = -1.5\cos(\dfrac{s}{8} - 3) + 5\) for \(s = 6.2\)
For Problems 35–38, find the reference angle in radians.
For Problems 39–40, find the angle of inclination of the line.
\(2x + 5y = -3\)
\(\dfrac{x}{8} - \dfrac{y}{11} = 1\)
For Problems 43–46,
For Problems 47–48, solve the equation graphically for \(0 \le x \lt 2\pi\text{.}\)
\(6 + \tan (x - \dfrac{\pi}{6}) = 7\)
\(3 - \tan (x + \dfrac{3\pi}{4}) = 4\)
For Problems 49–52, solve the equation exactly for \(0 \le x \lt 2\pi\text{.}\)
\(\sin \theta = \dfrac{\sqrt{3}}{2}\)
\(\sin \theta = -\dfrac{1}{2}\)
\(\cos \theta = -1\)
\(\tan \theta = \sqrt{3}\)
For Problems 53–58, find all solutions between \(0\) and \(2\pi\text{.}\) Round to two decimal places.
\(\tan t = 5\)
\(\cos x = -0.63\)
\(\sin h = -0.26\)
\(\tan \phi = -2.5\)
\(\cos \beta = 0.95\)
\(\sin \alpha = 0.1\)
For Problems 59–62, solve for \(x\text{.}\)
For Problems 63–66, sketch the graph. State its domain and range.
\(g(x) = 2x^2 + 4\)
\(h(w) = 1 - \dfrac{1}{w^2}\)
\(F(s) = -\sqrt{16 - s^2}\)
\(G(t) = 4 + \sqrt{4 - t}\)
Prove the Pythagorean identity \(\cos^2 t + \sin^2 t = 1\) by carrying out the following steps. Sketch a unit circle, and an arc \(t\) in standard position.
Prove the tangent identity \(\tan t = \dfrac{\sin t}{\cos t}\) by carrying out the following steps. Sketch an arc \(t\) in standard position on a unit circle, and label its terminal point \((x,y)\text{.}\)