Trigonometric Functions: Learn It 1

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Trigonometric Functions: Learn It 1

Switch between degree and radian measurements for angles
Understand and use the basic rules and relationships in trigonometry
Analyze trigonometric functions by examining their graphs, identifying cycles, and describing shifts in sine and cosine graphs

Degrees versus Radians

Probably the most familiar unit of angle measurement is the degree. One degree is $\frac{1}{360}$ of a circular rotation, so a complete circular rotation contains $360$ degrees. An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol $°$ .

You will use degrees a lot as you learn geometry but you may see it in other areas of life as well. Degrees are used in:

Navigation – navigation systems, such as compasses and GPS devices, use degrees to indicate directions.
Construction – degrees are used in construction and engineering to measure and specify angles when building structures. Architects, carpenters, and engineers use degrees to determine the angle of roof slopes, the inclination of ramps, or the angles of intersecting beams.
Astronomy – astronomers use degrees to describe the positions of celestial objects, angular separations between stars or planets, and the size of apparent motions of celestial bodies
Sports – in golf, angles are used to calculate the direction and trajectory of shots. In basketball, the angle of a player’s jump shot can affect the ball’s path to the basket
Art and design – when creating perspective drawings or determining the tilt and angles of lines in graphic design, degrees are used to ensure accurate proportions and compositions

Radians provide an alternative to degrees for measuring angles and are often preferred in mathematics because they have a natural relationship with circle geometry. Radians are based on the concept of using the radius of a circle to measure angles.

One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals $2π$ times the radius, a full circular rotation is 2 $π$ radians. The symbol for radians is $\text{rad}$ .

Pi ( $π$ ) is a mathematical constant approximately equal to $3.14$ . It represents the ratio of a circle’s circumference to its diameter.

Since there are $360$ degrees in a circle and $2π$ radians in a circle, the conversion factor between degrees and radians is $\frac{180}{π}$ . It is expected to keep the $π$ symbol when discussing radians, not converting to decimals, in order to maintain precision.

\begin{array}{rcl} 2 π radians & = & 360^{\circ} \\ π radians & = & \frac{360^{\circ}}{2} = 180^{\circ} \\ 1 radian & = & \frac{180^{\circ}}{π} \approx {57.3}^{\circ} \end{array}

$\begin{array}{rcl} 2\pi \text{ radians} & = & 360^\circ \\ \pi \text{ radians} & = & \frac{360^\circ}{2} = 180^\circ \\ 1 \text{ radian} & = & \frac{180^\circ}{\pi} \approx 57.3^\circ \end{array}$

degrees versus radians

Degrees are the most common measurement of angles. A circle is divided into $360^\circ$ . The symbol for degrees is a small, raised circle: $^\circ$ .
Radians are an alternative unit of angle measurement. In a circle, there are $2π$ radians. The symbol for radians is $\text{rad}$ .

The conversion factor between degrees and radians is $\frac{180}{π}$ .

$\text{Angle in Degrees} = \text{Angle in Radians }\times\frac{180}{π}$

$\text{Angle in Radians} = \text{Angle in Degrees }\times\frac{π}{180}$

Degrees are commonly used in everyday situations like navigation, construction, and basic geometry, while radians are more prevalent in advanced mathematics, physics, engineering, and other scientific fields.

Express $225°$ using radians.
Express $\dfrac{5\pi}{3}$ rad using degrees.

Use the fact that $180°$ is equivalent to $\pi$ radians as a conversion factor: $1=\dfrac{\pi \, \text{rad}}{180^{\circ}}=\dfrac{180^{\circ}}{\pi \, \text{rad}}$ .

$225^{\circ}=225^{\circ}·\dfrac{\pi }{180^{\circ}}=\dfrac{5\pi }{4}\text{ rad}$
$\dfrac{5\pi }{3}\text{ rad}$ = $\dfrac{5\pi }{3}·\dfrac{180^{\circ}}{\pi }=300^{\circ}$

A Ferris wheel with a radius of $20$ meters makes one complete rotation every $5$ minutes. A passenger boards at the bottom of the wheel. After $1$ minute, what is the angle of rotation in:

degrees?
radians?

Angle of rotation after $1$ minute:

Full rotation takes $5$ minutes
In $1$ minute, it rotates $\frac{1}{5}$ of a full circle

In degrees: $(\frac{1}{5}) × 360° = 72°$
In radians: $(\frac{1}{5}) × 2π \text{ rad} = \frac{2π}{5} rad ≈ 1.26 \text{ rad}$