Warm Up Find the inverse in expanded form: f x =−4+ 𝑥−5 8 Solve the system: 𝑥 2 + 𝑦 2 =7 5𝑥 2 − 𝑦 2 =1 3) Factor: 5𝑥 2 +3𝑥−8 4) Simplify: 5 2𝑥−3 2

1 revolution = 360 degrees = 2 radians Fill in each unit circle with the degree and radian measure for each line.

Final Exam Average Median 6th 83.8 87 7th 83.7 85.5 8th 85 89.5

Section 7-1 Measurement of Angles Chapter 7 Trigonometric Functions Section 7-1 Measurement of Angles Objective: To find the measure of an angle in either degrees or radians.

Common Terms Initial ray - the ray that an angle starts from. Terminal ray - the ray that an angle ends on. Vertex – the starting point A revolution is one complete circular motion.

Standard Position of an Angle The vertex of the angle is at (0,0). Initial ray starts on the positive x-axis. The terminal ray can be in any of the quadrants. The initial side is located on the positive x-axis The vertex is at origin

The angle describes the amount and direction of rotation. When sketching angles, always use an arrow to show direction. –210° 120° Positive Angle: rotates counter-clockwise (CCW) Negative Angle: rotates clockwise (CW)

Units of Angle Measurement Degree 1/360th of a circle. This is the measure on a protractor and most people are familiar with.

Units of Angle Measurement Radian Use the string provided to measure the radius. Start on the x-axis and use the string to measure an arc the same length on the circle. The angle created is one radian.

When the arc of circle has the same length as Angle θ is one radian Arc Length = Radius When the arc of circle has the same length as the radius of the circle, angle  measures 1 radian.

Units of Angle Measurement Radian Use the string provided to show an angle of 2 radians. How many radians make a complete circle?

Units of Angle Measurement Radian Use the string provided to show an angle of 2 radians. How many radians make a complete circle?

To convert degrees to radians, multiply by 𝝅 𝟏𝟖𝟎 To convert radians to degrees, multiply by 𝟏𝟖𝟎 𝝅 Convert 196˚ to radians. 196˚∗ 𝜋 180˚ = 196𝜋 180 = 49𝜋 45 radians Convert 1.35 radians to degrees. 1.35∗ 180˚ 𝜋 =77.35˚

Section 7-2 Sectors of Circles Objective: To find the arc length and area of a sector of a circle and to solve problems involving apparent size.

Sector of a Circle K= area of the sector A sector of a circle is the region bounded by a central angle and the intercepted arc. s = arc length 𝐴𝐵 𝜃= central angle r = radius K= area of the sector

Degrees Radians 𝑠= 𝜃𝑟𝜋 180 𝑠=𝑟 𝐾= 1 2 𝑟2 𝐾= 𝜃 𝑟 2 𝜋 360 𝐾= 1 2 𝑟𝑠 𝑠= 𝜃𝑟𝜋 180 𝑠=𝑟 𝐾= 1 2 𝑟2 𝐾= 𝜃 𝑟 2 𝜋 360 𝐾= 1 2 𝑟𝑠 s = arc length 𝜃= central angle r = radius K= area of the sector

Find the arc length and area of each sector.

𝐾= 1 2 𝑟 2 𝜃 𝑠=𝜃𝑟 𝑠= 2𝜋 3 ∗6=4𝜋 in Arc Length: Area: 𝐾= 1 2 𝑟 2 𝜃 Area: 𝐾= 1 2 ∗ 6 2 ∗ 2𝜋 3 =12𝜋 𝑖𝑛 2

𝑠= 45∗4∗𝜋 180 =𝜋 cm Arc Length: Area: 𝐾= 45∗ 4 2 ∗𝜋 360 =2𝜋 𝑐𝑚 2 𝒔= 𝜽𝒓𝝅 𝟏𝟖𝟎 𝑠= 45∗4∗𝜋 180 =𝜋 cm 𝑲= 𝜽 𝒓 𝟐 𝝅 𝟑𝟔𝟎 Area: 𝐾= 45∗ 4 2 ∗𝜋 360 =2𝜋 𝑐𝑚 2

Apparent Size 𝑠= 𝜃𝑟𝜋 180 𝑠=𝑟 How big an object looks depends not only on its size but also on the angle that it subtends at our eyes. The measure of this angle is called the object’s apparent size. s r 𝑠= 𝜃𝑟𝜋 180 𝑠=𝑟

Jupiter has an apparent size of 0.01° when it is 8 x 108 km from Earth. Find the approximate diameter of Jupiter. 𝑠= 𝜃𝑟𝜋 180 𝑠= 8× 10 8 .01 𝜋 180 =139,626 km

Homework Page 261 #1-11 odds Page 264 #1-17 odds