Lesson: Trigonometric Ratios on the Unit Circle

In this lesson, we will learn how to use the fact that the quadrant where an angle lies determines the signs of its sine, cosine, and tangent and solve trigonometric equations.

Sample Question Videos

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  • 02:04
  • 02:47

Worksheet: Trigonometric Ratios on the Unit Circle • 14 Questions • 6 Videos

Q1:

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

Q3:

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

Q4:

Is c o s 4 0 0 ∘ positive or negative?

Q5:

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

Q6:

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

Q7:

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

Q8:

Which function does the plot shown in the graph represent?

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

Q9:

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

Q10:

In the figure, points 𝑀 ( 𝜃 , 𝜃 ) c o s s i n 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 .

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 135?

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?

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

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

Q13:

Consider the following figures.

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

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

Q14:

Which function does the plot shown in the graph represent?

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