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 s i n 𝜃 , given 𝜃 is in standard position and its terminal side passes through the point 3 5 , 4 5 .

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

Q2:

Find t a n 𝜃 given 𝜃 is in standard position and its terminal side passes through the point 3 5 , 4 5 .

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

Q3:

Determine the quadrant in which 𝜃 lies if c o s 𝜃 < 0 and s i n 𝜃 > 0 .

  • Athe third quadrant
  • Bthe first quadrant
  • Cthe fourth quadrant
  • Dthe second quadrant

Q4:

Is c o s 4 0 0 positive or negative?

  • Apositive
  • Bnegative

Q5:

Find s e c 𝜃 , given 𝜃 is in standard position and its terminal side passes through the point 4 5 , 3 5 .

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

Q6:

In which quadrant does 𝜃 lie if s i n 𝜃 = 1 2 and c o s 𝜃 = 1 2 ?

  • A the third
  • B the second
  • C the fourth
  • D the first

Q7:

Determine the quadrant in which 𝜃 lies if c o s 𝜃 > 0 and s i n 𝜃 > 0 .

  • Athe third quadrant
  • Bthe second quadrant
  • Cthe fourth quadrant
  • Dthe first quadrant

Q8:

Which function does the plot shown in the graph represent?

  • Atangent
  • Bcosine
  • Csine

Assign each region of the plot with the corresponding quadrant of the unit circle.

  • AA: IV, B: I, C: II, D: III
  • BA: III, B: IV, C: I, D: II
  • CA: II, B: III, C: IV, D: I
  • DA: I, B: II, C: III, D: IV
  • EA: I, B: II, C: IV, D: III

Q9:

In which quadrant does 𝜃 lie if s i n 𝜃 > 0 and c o s 𝜃 > 0 ?

  • Athird
  • Bsecond
  • Cfourth
  • Dfirst

Q10:

In the figure, points 𝑀 ( c o s 𝜃 , s i n 𝜃 ) and 𝑁 lie on the unit circle, and 𝐴 𝑂 𝑁 = 2 𝜋 𝜃 .

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

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

Q11:

If the angle 𝜃 is in the standard position, c o s 𝜃 = 2 2 , and s i n 𝜃 = 2 2 , is it possible for 𝜃 to measure 1 3 5 ?

  • ANo
  • BYes

Q12:

A point 𝑅 is traveling on a unit circle at a speed of one complete rotation per second.

If its position represents an angle 𝜃 at a time 𝑡 , when will it have the same position again?

  • Ahalf a second later
  • Bone-third of a second later
  • Cone and a half seconds later
  • Done second later
  • Etwo seconds later

How would you express the angle representing its position at this time?

  • A 𝜃 + 2 𝜋
  • B 𝜃 + 𝜋 3
  • C 𝜃 + 𝜋 2
  • D 𝜃 + 𝜋
  • E 𝜃 + 𝜋 4

Deduce what the period of the cosine and sine functions is.

  • A 𝜋 4
  • B 𝜋
  • C 2 𝜋
  • D 𝜋 2
  • E 𝜋 3

Q13:

Consider the following figures.

Which function does the plot in the graph, figure (a), represent?

  • Atangent
  • Bsine
  • Ccosine

Assign each region of the plot in figure (a) with the corresponding quadrant of the unit circle in figure (b).

  • AA: II, B: III, C: IV, D: I
  • BA: III, B: IV, C: I, D: II
  • CA: I, B: II, C: III, D: IV
  • DA: IV, B: I, C: II, D: III
  • EA: I, B: II, C: IV, D: III

Q14:

Which function does the plot shown in the graph represent?

  • Atangent
  • Bcosine
  • Csine

Assign each region of the plot with the corresponding quadrant of the unit circle.

  • AA: III, B: IV, C: I, D: II
  • BA: II, B: III, C: IV, D: I
  • CA: IV, B: I, C: II, D: III
  • DA: I, B: II, C: III, D: IV
  • EA: I, B: II, C: IV, D: III

Q15:

The terminal side of 𝐴 𝑂 𝐵 in standard position intersects with the unit circle at the point 𝐵 with coordinates ( 0 . 8 , 0 . 6 ) . Find t a n 𝐴 𝑂 𝐵 .

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

Q16:

Find c o t 𝜃 given 𝜃 is in standard position and its terminal side passes through the point 8 1 7 , 1 5 1 7 .

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

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