Trigonometric Definitions Happy Monday

Definitions Angle : AOB consists of two rays 𝑅 1 and R 2 with a common vertex O. We often interpret angels as a rotation of the ray 𝑅 1 𝑜𝑛𝑡𝑜 𝑅 2 .

Definitions Initial side: 𝑅 1 Terminal Side: 𝑅 2 If the rotation is counterclockwise the angle is considered positive If the rotation is clockwise the angle is considered Negative

Definitions Measure: the amount of rotation about the vertex required to move 𝑅 1 𝑜𝑛𝑡𝑜 𝑅 2 . This is how much the angle “opens”

Definitions One unit of measurement for angles is degree. Degree: an angle of measure 1 degree is formed by rotating the initial side 1 360 of a complete revolution.

Definitions Formal Definition Radian: If a circle of radius 1 is drawn with the vertex of an angle at its center, then the measure of this angle in radians (rad) is the length of the arc that subtends the angle.

Definitions Less formal Radian: The amount an angle opens is measured along the arc of circle of radius 1, with the center at the vertex of the angle.

Angles in a Circle Degrees in any circle is 360° Radians: The circumference of the circle of radius 1 is 2𝜋 and so a complete revolution has measure 2𝜋 𝑟𝑎𝑑 An angle that is subtended by an arc length 2 along the unit circle has a radian measure 2.

Conversions Degrees to Radians: Multiply by 𝜋 180 Radians to Degrees: Multiply by 180 𝜋

Definitions Standard position: It is in standard position if it is drawn in the 𝑥𝑦 plane with its vertex at the origin and its initial side on the positive x-axis.

Definitions Coterminal: Two angles are considered co-terminal if the angles coincide. Basically add or subtract 360° 𝑜𝑟 2𝜋 to any angle and it will be co-terminal.

On back of the paper Sketch 2 separate angles, both in standard position, one angle should be positive the other negative. Estimate the angle measure of the angles you drew. From your estimation find two co-terminal angles.

Definitions Length of a circular arc: In a circle with radius 1 the length s of an arc that subtends a central angle of 𝜃 radians is 𝑠=𝑟𝜃

Area of Circular Sector: Area of Circular Sector: Area of a circle is 𝐴=𝜋 𝑟 2 . A sector of this circle with central angle 𝜃 has an area that is the fraction 𝜃 2𝜋 of the entire circle. So 𝜋 𝑟 2 ∗ 𝜃 2𝜋 = 𝜋𝑟 2 𝜃 2𝜋 = 𝑟 2 𝜃 2 or 1 2 𝑟 2 𝜃

Practice: Find the radian measure of the angle with the given degree: 50° 2) 300° 3) 65° 4) −150° Find the degree measure of the angle given in radian measure 5) 3𝜋 4 6) 5𝜋 6 7) 1.5 8) 𝜋 18

Answers 5𝜋 18 5𝜋 3 13𝜋 36 − 5𝜋 13 5) 135° 6) 150° 7) 270 𝜋 ° 8) 10°

Find two positive and two negative angels that are conterminal 360°, 750° , −330°, −690° 510°,870°, −210°,−570° 290°,650°,−430°,−790° 3𝜋, 5𝜋, −𝜋,−3𝜋 11𝜋 4 , 19𝜋 4 , − 5𝜋 4 , − 13𝜋 4 3𝜋 2 , 5𝜋 2 , − 5𝜋 2 , − 9𝜋 4 30° 150° −70° 4) 𝜋 5) 3𝜋 4 6) − 𝜋 2

Calculate the arc length and sector area in terms of 𝜋 of the following 𝑟=1, 𝜃=𝜋 𝑟=1, 𝜃= 𝜋 2 𝑟=2,𝜃= 𝜋 4 𝑟=4,𝜃=6𝜋 1) 𝑠=𝜋 𝐴= 1 2 𝜋 or 𝜋 2 2) 𝑠= 𝜋 2 𝐴= 𝜋 4 3) 𝑠= 𝜋 2 𝐴= 𝜋 2 4) 𝑠=24𝜋 𝐴=48𝜋

Homework Pages 453-455 1st page in packet #’s 19-42 𝒐𝒅𝒅 𝟒𝟑, 𝟒𝟒, 𝟒𝟓, 𝟓𝟖