6.2 Radian Measure and angles on the Cartesian Plane June 2020 PDF

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Math lecture notes, all filled in and ready to study. recommend using for the midterm, but wont be on the final....

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MHF 4U

6.2 Radian Measure and Angles on the Cartesian Plane

Learning Goal: Use the Cartesian plane to evaluate the trigonometric ratios for angles between, 0 and 2 Angles in Standard Position: An angle  is in standard position if the vertex of the angle is at the origin and the initial arm lies along the positive x-axis. The terminal arm can be anywhere on the arc of rotation. The principal angle is the angle between 0 and 2 ( i.e. 0 and 360 ) The related acute angle is the angle formed by the terminal arm of an angle in standard position and the x-axis. The related acute angle is always positive and lies between 0 and (i.e 0 and 90 )

Positive and Negative Angles:

The CAST Rule: The CAST rule is a way to remember whether a trig ratio is positive or negative in a given quadrant. C - Cosine is positive in the fourth quadrant. A - All trig ratios are positive in the first quadrant. S - Sine is positive in the second quadrant. T - Tan is positive in the third quadrant.

 2

Example 1: Sketch the angle in standard position and find the related acute angle. a)  =

2 3

b)  =

7 6

c) 𝜃 = −

3𝜋 4

The angles in the special triangles can be expressed in radians, as well as in degrees. The radian measure can be used to determine the exact values of the trigonometric ratios for multiples of these angles between 0 and 2 . Therefore, the special triangles using radians are as follows:

Therefore, the exact values of the three primary trigonometric ratios and their reciprocals (secondary trigonometric ratios) can be determined. Recall: secondary trigonometric ratios

** Note: It is the trigonometric ratio that’s used for the reciprocal, NOT the angle itself.

Example 2: Determine the exact value of the following trigonometric ratios. a)

sin

3𝜋 4

b)

sec

5𝜋 6

c) cot

3 2

Example 3: State an equivalent expression in terms of the related acute angle. a)

cos

5 4

b)

  tan  −   4

The same strategies can be used to determine the values of trigonometric ratios on a Cartesian plane in radians as in degrees. The trigonometric ratios for any principal angle,  , in standard position can be determined by finding the related acute angle,  , using the coordinates of any point that lies on the terminal arm of the angle.

Example 4: For the following, determine the radian value of  if 0    2 𝑡𝑎𝑛𝜃 =

−7 24

Example 5: The terminal arm of an angle in standard position passes through the point ( - 4, -2). Determine the radian value of the angle in the interval 0,2  , to the nearest hundredth.

Practice: Pg. 330 # 1a-d,2ab,3,4,5c-f,6a-c,7a-c,8a-c,13ac           Corrections: 4c) − cot  4d) − sec  8a) − cos  8b) − tan   8c) − csc  4  6   4  6 3 ...