Lesson Worksheet: Trigonometric Ratios on the Unit Circle Mathematics • 10th Grade

In this worksheet, we will practice relating the 𝑥- and 𝑦-coordinates of points on the unit circle to trigonometric functions.

Q1:

Find the value of cos0.

Q2:

In the figure, points 𝑀(πœƒ,πœƒ)cossin and 𝑁 lie on the unit circle, and βˆ π΄π‘‚π‘=2πœ‹βˆ’πœƒ.

Express the values of sine, cosine, and tangent of 2πœ‹βˆ’πœƒ in terms of their values for πœƒ. Check whether this is valid for all values of πœƒ.

  • Acoscossinsintantan(2πœ‹βˆ’πœƒ)=βˆ’πœƒ,(2πœ‹βˆ’πœƒ)=βˆ’πœƒ,(2πœ‹βˆ’πœƒ)=βˆ’πœƒ
  • Bcoscossinsintantan(2πœ‹βˆ’πœƒ)=πœƒ,(2πœ‹βˆ’πœƒ)=βˆ’πœƒ,(2πœ‹βˆ’πœƒ)=βˆ’πœƒ
  • Ccoscossinsintantan(2πœ‹βˆ’πœƒ)=πœƒ,(2πœ‹βˆ’πœƒ)=βˆ’πœƒ,(2πœ‹βˆ’πœƒ)=πœƒ
  • Dcoscossinsintantan(2πœ‹βˆ’πœƒ)=πœƒ,(2πœ‹βˆ’πœƒ)=πœƒ,(2πœ‹βˆ’πœƒ)=βˆ’πœƒ
  • Ecoscossinsintantan(2πœ‹βˆ’πœƒ)=βˆ’πœƒ,(2πœ‹βˆ’πœƒ)=πœƒ,(2πœ‹βˆ’πœƒ)=πœƒ

Q3:

Find sinπœƒ, given πœƒ is in standard position and its terminal side passes through the point ο€Ό35,βˆ’45.

  • A45
  • B35
  • Cβˆ’35
  • Dβˆ’45

Q4:

Find secπœƒ, given πœƒ is in standard position and its terminal side passes through the point ο€Ό45,35.

  • A53
  • B54
  • C45
  • D35

Q5:

Find cot(180+πœƒ)∘ given πœƒ is in standard position and its terminal side passes through the point ο€Όβˆ’2129,βˆ’2029.

  • Aβˆ’2021
  • B2021
  • Cβˆ’2120
  • D2120

Q6:

The terminal side of βˆ π΄π‘‚π΅ in standard position intersects with the unit circle at the point 𝐡 with coordinates (βˆ’π‘₯,βˆ’π‘₯) where π‘₯ is a postive number. Find sinπœƒ.

  • Aβˆ’1√2
  • B1
  • C1√2
  • Dβˆ’1

Q7:

The terminal side of βˆ π΄π‘‚π΅ in standard position intersects the unit circle 𝑂 at the point 𝐡 with coordinates ο€Ώ3√10,𝑦, where 𝑦>0. Find the value of sin𝐴𝑂𝐡.

  • A110
  • B13
  • C3√10
  • D1√10

Q8:

Suppose 𝑃 is a point on a unit circle corresponding to the angle of 4πœ‹3. Is there another point on the unit circle representing an angle in the interval [0,2πœ‹) that has the same tangent value? If yes, give the angle.

  • AYes, 11πœ‹6
  • BNo
  • CYes, πœ‹4
  • DYes, πœ‹3
  • EYes, πœ‹6

Q9:

Consider 𝐴, a point on a unit circle corresponding to the angle of 3πœ‹2. Is there another point on the unit circle that has the same 𝑦-coordinate as 𝐴 and represents an angle in the interval [0,2πœ‹)? If yes, give the angle.

  • AYes, πœ‹4
  • BYes, πœ‹3
  • CYes, πœ‹6
  • DNo
  • EYes, πœ‹2

Q10:

Find the equation of the straight line that makes an angle of 3πœ‹4, measured in radians, with the positive direction of the π‘₯-axis in the standard position in the unit circle.

  • A𝑦=βˆ’π‘₯
  • B𝑦=π‘₯
  • C𝑦=√3π‘₯3
  • D𝑦=βˆ’βˆš3π‘₯3

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