Lesson 6.3 ~ Radian Measure Objectives: • I understand the definition of a radian. • I can convert between radian and degree measure. • I can find the sine and cosine of a radian measure. • I can find arc length in radian measure.

Page 395-402

Not in book

A RADIAN is the measure of a central angle that intercepts an arc with the length equal to the radius of the circle. Radians measure the amount of rotation between the initial side and terminal side of an angle (it’s another way to measure an angle). An angle with a full circle rotation measures 2𝜋 radians (which means that 360° = 2𝜋 radians). An angle with a half circle rotation measures 𝜋 radians (which means that 180° = 𝜋 radians). Since we will use both degree measure and radian measure, it is VERY important that we use correct notation. If an angle measure is in degrees, you MUST use the degree (°) symbol. There isn’t a symbol for radians, so if we see an angle measure without a degree symbol, we always assume that it is being measured in radians.

Page 395

If the denominator on the left was 360°, what value would be in the denominator on the right?

Why does it work? Because the circumference of the circle is equal to 2𝜋𝑟, that means that there are always 2𝜋 radians in any circle. Since every circle also measures 360°, than that means 2𝜋 = 360°, and it then follows that 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 180°.

Page 395

How do you remember which fraction to multiply by when converting between radians and degrees? The original units should always cancel when you multiply.

Page 396

Not in book

(page 396)

Note that part “C” doesn’t have 𝜋 in it, even though it’s in radians. Can you estimate a reasonable answer so you can check your answer?

Remember the definition for “Coterminal Angles”? We can find coterminal angles in radians in the same way we do with degree measures. With angles measured in degrees, we added and subtracted 360°. What do we need to add and subtract when dealing with radians? Can we always use that value, or will we sometimes need to change it’s appearance? (Think

about the case where the angle we are looking at is 𝜋

2.)

Not in book

Example: Find one positive and one negative coterminal angle measure for each of the angles below.

a) 3𝜋

4 radians b)

13𝜋

6 radians

c) −𝜋

2 radians c) −

15𝜋

7 radians

Not in book

Not in book

Journal Entry: what is cosθ?

(page 397)

Stop and think! An angle in standard position on a unit circle can have a negative measure. When calculating the length of an intercepted arc, we take the absolute value of the angle measure? Why?

Page 398

Could we find the remaining arc length of the circle above? How?

Not in book

(page 398-399)

three decimal places.

(page 398-399)

three decimal places.

Landing our Big Ideas:

• 2π radians around any circle Radians:

• Convert Radians to degrees, degrees to radians

Find sine and cosine: cos is x coordinate, sin is y.

Find arc length in radians. Arc length and arc measure are NOT the same.

Assignment 6.3: pg. 396 – 402 # 1 – 6, 8 – 11, 13, 19 – 22 *On questions 19 – 22 only find the values of the cosine and sine of the angle