Learning objective
Angles are defined as geometric objects - wherever two lines, segments or rays intersect. We can also define an angle as the action of rotating a ray about its endpoint.
Consider the angles shown.
Which angle is in standard position?
Determine whether the angle in standard position has a positive or negative measure.
An angle is in standard position when its vertex is at the origin and its initial side coincides with the positive xaxis.
The terminal side, which is free to rotate, determines the angle's measure in either positive (counterclockwise) or negative (clockwise) directions.
Drag the sliders to explore the applet. The different colors show the different lengths of each radian measure around the arc of the circle.
Recall that an angle in radians can be calculated as \theta = \dfrac{s}{r}, where s is the arc length and r is the radius of the circle.
If we now restrict our attention to circles of radius one unit, then 1 radian would be the angle subtended by an arc of length 1 unit.
The circumference of a circle of radius one unit is C= 2 \pi \left( 1 \right) = 2 \pi, so the angle represented by a full turn around a circle is 2 \pi radians =6.28 radians =360 \degree.
Since a half-circle is equivalent to \pi radians, and \pi = 180 \degree, angles given in radian measure are commonly expressed as fractions of \pi.
For each fraction of a complete circle shown in the table below, find the corresponding degree and radian measures of each angle. If possible, simplify the radian degree measures.
1 | \dfrac{3}{4} | \dfrac{2}{3} | \dfrac{1}{2} | \dfrac{1}{3} | \dfrac{1}{4} | \dfrac{1}{6} | \dfrac{1}{8} | \dfrac{1}{12} | |
---|---|---|---|---|---|---|---|---|---|
Measure in degrees | 360 \degree | ||||||||
Measure in radians |
For each of the following radian measures, sketch and label the corresponding angle on the unit circle in standard position. Explain how you chose to place the terminal side of the angle.
\dfrac{\pi}{2}
\dfrac{3 \pi}{2}
\dfrac{\pi}{3}
\dfrac{4 \pi}{3}
Find the missing measure for each circle. Round your answers to two decimal places.
Given a circle with a 12 \text{ ft} arc subtended by the angle \dfrac{\pi}{4}, find the radius.
Given a circle with a radius of 9 \text{ m}, find the measure of the arc subtended by a 2.09 \text{ radian} angle.
Given a circle with a radius of 6 \text{ cm}, find the radian measure of the angle subtended by a 33 \text{ cm} arc.
A radian is the measure of an angle \theta that, when drawn as a central angle, subtends an arc whose length equals the length of the radius of the circle.
We use \pi = 180 \degree and equivalent fractions of a unit circle to convert between degrees and radians.
It is also possible to have an angle that rotates more than once around the circle. Rotations of this type will have measures with a magnitude greater than 2 \pi.
Because of the rotation definition of an angle, it's possible to have two angles with the same initial and terminal sides but different measures. Angles that are related in this way are called coterminal angles.
In general, an angle coterminal with another angle differs from it by an integer multiple of 2 \pi.
For the unit circle, we measure angles of any magnitude between the positive x-axis and the radius to a point that moves on the circle. The trigonometric functions of those angles are defined in a manner that guarantees that a function of any angle will be related to the same function of an angle in the first quadrant.
We often use \theta for the angle and \alpha for the acute reference angle between 0 and \dfrac{\pi}{2} in the first quadrant.
To find a reference angle, first, if necessary, add or subtract multiples of 2 \pi to obtain an angle between 0 and 2 \pi. Then, use the symmetry of the unit circle to find the size of the acute angle in relation to the x-axis.
Consider the angle - \dfrac{5 \pi}{4}.
Graph the angle in standard form.
Find the measure of the coterminal angle to - \dfrac{5 \pi}{4} that is between 0 and 2 \pi.
Find the measure of the reference angle for - \dfrac{5 \pi}{4}.
For each of the following angle measures, determine the coterminal angle between 0 and 2\pi and graph the angle in standard form on a coordinate plane.
\dfrac{35 \pi}{6} radians
-\dfrac{8 \pi}{3}
Coterminal angles have two angles with the same initial and terminal sides but different measures. To find coterminal angles, add or subtract an integer multiple of 2 \pi.
A reference angle is an acute angle formed by the terminal side of an angle and the x-axis.
To find a reference angle, first, if necessary, add or subtract multiples of 2 \pi to obtain an angle between 0 and 2 \pi. Then, use the symmetry of the unit circle to find the size of the acute angle in relation to the x-axis.
The image below shows a point P \left(x, y \right) with a terminal side length r.
Once we place trigonometric ratios on the coordinate plane, we can now define trigonometric functions with respect to the acute reference angle \theta as:\begin{aligned} \text{The sine ratio: } \sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}} & = \dfrac{y}{r} \\\\ \text{The cosine ratio: } \cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}} & = \dfrac{x}{r} \\\\ \text{The tangent ratio: } \tan \theta = \dfrac{\text{opposite}}{\text{adjacent}} & = \dfrac{y}{x} \end{aligned}
Counterclockwise rotations of the terminal side of the angle represent positive angle measures, while clockwise rotations represent negative angle measures.
Angles may be measured in degrees or radians, and may be rational or irrational numbers. Angles larger than 360 \degree or 2 \pi result from continuing to rotate around the circle in multiple rotations. Thus, we can extend the definition of trigonometric functions to include values for all angles \theta in the set of real numbers.
Explore the applet by dragging the triangle and checking the box.
The point on the following graph has coordinates \left( -7, -24 \right).
Find r, the distance from the point to the origin.
Evaluate the sine, cosine, and tagent functions for \theta. Leave the values of the functions as ratios.
The sine ratio is represented by the following.
The cosine ratio is represented by the following.
The tangent ratio is represented by the following.