I can use both Radians and Degrees to Measure Angles

What is trigonometry? This is one of two chapters on trigonometry. Make sure that you get off to a good start! Originally trigonometry dealt only with triangle relationships, we will now expand it to apply to a variety of contexts.

What is an angle? An angle is formed by rotating a ray around its endpoint. Each angle has an initial side, vertex, and terminal side Placing the vertex at the origin and the initial side along the x-axis puts the angle in standard position Positive angles are produced by counterclockwise rotation and negative angles are produced by clockwise rotation

What is a radian? The measure of an angle is determined by the amount of rotation. One helpful way of measuring an angle is to use radian measure. To define a radian, you can use the central angle of a circle. (This means the vertex is at the center.) One radian is the measure of a central angle, θ, that intercepts an arc, s, equal in length to the radius of the circle, r Draw a picture of this and try to estimate the number of degrees in one radian.

In general the radian measure of an angle θ = arc length = s radius r (Also, arc length, s = r θ ) Because the circumference of a circle is 2πr, then one full rotation corresponds to an arc length of 2π r. So, how many radians is one full rotation?

So ½ revolution = π radians ¼ revolution = π/2 radians 1/6 revolution = π/3 radians Can you use the last statement to estimate the degree measure of a radian? Radians are really a ratio, thus they have no unit associated with them. It will be very important to start to “think” in radians!

Coterminal Angles Angles are coterminal if they share the same terminal ray when in standard position. Therefore π, 3π, and –π are coterminal. In general you can add or subtract multiples of 2π or 360 degrees. Find three angles coterminal with π/4.

Complements and Supplements What are complementary angles? (Explain using radian measure.) What are supplementary angles? We will only use positive angles in these definitions. Find the complement and supplement of π/6. (Notice that I did not need to say “radians”. It is assumed.)

Degree Measure Because one full revolution is 2π radians, degrees and radians are related by the formula 360˚= 2π radians So 1˚= π/180 radians and 1 radian = (180/π)˚

Conversions between degrees and radians Degrees to radians multiply by π/180˚ Radians to degrees multiply by 180˚/π. Change 135˚, 540˚, and -270˚to radians. Change –π/2, 9π/2, and 2 to degrees.

The diameter of a circle is 10 inches. A central angle cuts off an arc 1 foot long. What is the radian measure of the angle? In a circle with a radius of 8 inches, how long is the arc cut off by a central angle which measures π/4?

Minutes and Seconds Instead of using decimals for fractions of degrees, we often use minutes and seconds. 1 minute = 1’ = 1/60 of 1° 1 second = 1” = 1/60 of 1 minute = 1/3600 of 1° See example 1 on page 232. A graphing calculator can also be used.

Let’s analyze the linear speed of an object moving along a circular path (at a constant rate): Linear Speed v = distance = s time t (Remember s = r θ ) Example: The second hand of a clock is 10.2 cm long. Find the linear speed of the tip of it in cm/sec.

Angular speed (how fast something is turning) Angular speed ω= θ t A 30 inch (diameter)bicycle tire makes one revolution every ½ second. Find its angular speed in radians per second. Find the linear speed the bike is travelling in inches per second. (and miles per hour?) Can you find a formula for the relationship between v and ω?

You can use proportions to find the area of sectors. Find the area of a sector with a central angle of 45 degrees and a diameter of 16 inches. Find the area of a sector with a central angle of π/3 and a radius of 9 inches.

Can you find a formula for the area of a sector without converting radian measure to degrees? A = 1/2r 2 θ