Worksheet: Trigonometric Ratios on the Unit Circle

In this worksheet, we will practice using the fact that the quadrant where an angle lies determines the signs of its sine, cosine, and tangent and solve trigonometric equations.

Q1:

Find sin𝜃, given 𝜃 is in standard position and its terminal side passes through the point 35,45.

  • A 4 5
  • B 3 5
  • C 3 5
  • D 4 5

Q2:

Find tan𝜃 given 𝜃 is in standard position and its terminal side passes through the point 35,45.

  • A 3 4
  • B 4 3
  • C 3 4
  • D 4 3

Q3:

Find sec𝜃, given 𝜃 is in standard position and its terminal side passes through the point 45,35.

  • A 5 3
  • B 5 4
  • C 4 5
  • D 3 5

Q4:

In the figure, points 𝑀(𝜃,𝜃)cossin and 𝑁 lie on the unit circle, and 𝐴𝑂𝑁=2𝜋𝜃.

Express the values of sine, cosine, and tangent of 2𝜋𝜃 in terms of their values for 𝜃. Check whether this is valid for all values of 𝜃.

  • A c o s c o s s i n s i n t a n t a n ( 2 𝜋 𝜃 ) = 𝜃 , ( 2 𝜋 𝜃 ) = 𝜃 , ( 2 𝜋 𝜃 ) = 𝜃
  • B c o s c o s s i n s i n t a n t a n ( 2 𝜋 𝜃 ) = 𝜃 , ( 2 𝜋 𝜃 ) = 𝜃 , ( 2 𝜋 𝜃 ) = 𝜃
  • C c o s c o s s i n s i n t a n t a n ( 2 𝜋 𝜃 ) = 𝜃 , ( 2 𝜋 𝜃 ) = 𝜃 , ( 2 𝜋 𝜃 ) = 𝜃
  • D c o s c o s s i n s i n t a n t a n ( 2 𝜋 𝜃 ) = 𝜃 , ( 2 𝜋 𝜃 ) = 𝜃 , ( 2 𝜋 𝜃 ) = 𝜃
  • E c o s c o s s i n s i n t a n t a n ( 2 𝜋 𝜃 ) = 𝜃 , ( 2 𝜋 𝜃 ) = 𝜃 , ( 2 𝜋 𝜃 ) = 𝜃

Q5:

The terminal side of 𝐴𝑂𝐵 in standard position intersects with the unit circle at the point 𝐵 with coordinates (0.8,0.6). Find tan𝐴𝑂𝐵.

  • A 3 4
  • B 0 . 6
  • C 0 . 8
  • D 2 3
  • E 4 3

Q6:

Find cot𝜃 given 𝜃 is in standard position and its terminal side passes through the point 817,1517.

  • A 1 5 8
  • B 8 1 5
  • C 8 1 5
  • D 1 5 8

Q7:

Find the value of sin90.

Q8:

Find the value of sec180.

Q9:

Find the value of cot180.

  • A1
  • B 1
  • C0
  • DUndefined

Q10:

Find the value of csc90.

Q11:

Find the value of tan270.

  • A0
  • B1
  • C 1
  • DUndefined

Q12:

Find the value of cos0.

Q13:

Suppose 𝑃 is a point on a unit circle corresponding to the angle of 4𝜋3. Is there another point on the unit circle representing an angle in the interval [0,2𝜋) that has the same tangent value? If yes, give the angle.

  • Ayes, 11𝜋6
  • Bno
  • Cyes, 𝜋4
  • Dyes, 𝜋3
  • Eyes, 𝜋6

Q14:

Consider 𝐴, a point on a unit circle corresponding to the angle of 3𝜋2. 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)? If yes, give the angle.

  • AYes, 𝜋4
  • BYes, 𝜋3
  • CYes, 𝜋6
  • DNo
  • EYes, 𝜋2

Q15:

Suppose 𝐿 is a point on a unit circle corresponding to the angle of 𝜋3. Is there another point on the unit circle that represents an angle in the interval [0,2𝜋) and has the same 𝑥-coordinate as 𝐿? If yes, give the angle.

  • Ano
  • Byes, 𝜋6
  • Cyes, 7𝜋12
  • Dyes, 5𝜋3
  • Eyes, 2𝜋3

Q16:

Consider 𝑀, a point on a unit circle corresponding to the angle of 𝜋. Is there another point on the unit circle representing an angle in the interval [0,2𝜋) that has the same 𝑥-coordinate as 𝑀? If yes, give the angle.

  • Ayes, 𝜋6
  • Byes, 𝜋3
  • Cno
  • Dyes, 𝜋2
  • Eyes, 𝜋4

Q17:

Find the equation of the straight line that makes an angle of 3𝜋4, measured in radians, with the positive direction of the 𝑥-axis in the standard position in the unit circle.

  • A 𝑦 = 𝑥
  • B 𝑦 = 𝑥
  • C 𝑦 = 3 𝑥 3
  • D 𝑦 = 3 𝑥 3

Q18:

Consider that 𝑁 is a point on a unit circle corresponding to the angle of 5𝜋6. Is there another point on the unit circle that has the same 𝑦-coordinate as 𝑁 and represents an angle in the interval [0,2𝜋)? If yes, give the angle.

  • Ayes, 𝜋4
  • Bno
  • Cyes, 7𝜋6
  • Dyes, 𝜋6
  • Eyes, 𝜋3

Q19:

The terminal side of 𝐴𝑂𝐵 in standard position intersects the unit circle 𝑂 at the point 𝐵 with coordinates 310,𝑦, where 𝑦>0. Find the value of sin𝐴𝑂𝐵.

  • A 1 1 0
  • B 1 3
  • C 3 1 0
  • D 1 1 0

Q20:

The terminal side of 𝜃 intersects the unit circle at the point 817,𝑦 where 𝑦>0. Find the value of 𝜃 giving the answer to the nearest second and the value of 𝑦 as a fraction.

  • A 𝑚 𝜃 = 2 4 1 5 5 3 9 and 𝑦=1517
  • B 𝑚 𝜃 = 6 1 5 5 3 9 and 𝑦=1517
  • C 𝑚 𝜃 = 6 1 5 5 3 9 and 𝑦=1517
  • D 𝑚 𝜃 = 1 2 8 5 5 5 6 and 𝑦=815

Q21:

The terminal side of 𝜃 in standard position intersects with the unit circle at the point 𝐵 with coordinates 45,35. Find the value of 𝜃 giving the answer to the nearest second.

  • A 3 2 3 7 4 8
  • B 3 6 5 2 1 2
  • C 1 4 3 7 4 8
  • D 2 1 6 5 2 1 2

Q22:

In the given figure, point 𝑀 is on the unit circle and 𝜃 lies in the interval 0,𝜋2.

From the right triangle 𝑂𝑀𝑋, what ratio gives tan𝜃?

  • A 1 𝑋 𝑀
  • B 1 𝑂 𝑋
  • C 𝑂 𝑋 𝑋 𝑀
  • D 𝑋 𝑀 𝑂 𝑋
  • E 𝑋 𝑀 𝑂 𝑋

Which of the following is true concerning the triangles 𝑂𝑀𝑋 and 𝑂𝑇𝐴?

  • AThey are isosceles.
  • BThey are scalene.
  • CThey are congruent.
  • DThey are similar.
  • EThey are equilateral.

What is the scale factor of 𝑂𝑀𝑋 to 𝑂𝑇𝐴?

  • A s i n 𝜃
  • B c o s 𝜃
  • C 1 𝜃 t a n
  • D 1 𝜃 c o s
  • E 1 𝜃 s i n

Use your answers to the previous parts to determine the length of 𝐴𝑇 in terms of sin𝜃 and cos𝜃.

  • A c o s s i n 𝜃 𝜃
  • B s i n c o s 𝜃 𝜃
  • C 1 𝜃 c o s
  • D 1 𝜃 t a n
  • E c o s 𝜃

Q23:

Consider a windmill with blades of length 1 m. The position of the top 𝑃 of a given blade is given by its coordinates (𝑎,𝑏) which depend upon angle 𝜃 as shown.

Express 𝑎 and 𝑏 as functions of the measure of angle 𝜃 in radians.

  • A 𝑎 = 𝜃 , 𝑏 = 𝜃 c o s s i n
  • B 𝑎 = 𝜃 , 𝑏 = 𝜃 s i n c o s
  • C 𝑎 = 𝜃 , 𝑏 = 𝜃 t a n s i n
  • D 𝑎 = 𝜃 , 𝑏 = 𝜃 c o s t a n
  • E 𝑎 = 𝜃 , 𝑏 = 𝜃 c o s s i n

If the angle 𝜃 at a certain time is 5𝜋3, what will it be after the blade has completed half a rotation?

  • A 5 𝜋 3
  • B 4 𝜋 3
  • C 3 𝜋 5
  • D 3 𝜋 8
  • E 8 𝜋 3

Q24:

In the given figure, point 𝑀 is on the unit circle and 𝜃 lies in the interval 𝜋2,𝜋.

Is tan𝜃 positive or negative?

  • Apositive
  • Bnegative

Which of the following is true concerning the triangles 𝑂𝑋𝑀,𝑂𝐴𝑇, and 𝑂𝐶𝐴?

  • AThey are similar.
  • BThey are isosceles.
  • CThey are equilateral.
  • DThey are congruent.
  • EThey are scalene.

Q25:

The terminal side of angle 𝜃 in standard position intersects with the unit circle at the point, (2𝑎,3𝑎) where 0<𝜃<𝜋2. Find the value of 𝑎.

  • A 5
  • B 1 5
  • C 1 3
  • D 1 1 3
  • E 1 1 3

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