# 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 , given is in standard position and its terminal side passes through the point .

• A
• B
• C
• D

Q2:

Find given is in standard position and its terminal side passes through the point .

• A
• B
• C
• D

Q3:

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

• A
• B
• C
• D

Q4:

In the figure, points and lie on the unit circle, and .

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

• A
• B
• C
• D
• E

Q5:

The terminal side of in standard position intersects with the unit circle at the point with coordinates . Find .

• A
• B
• C
• D
• E

Q6:

Find given is in standard position and its terminal side passes through the point .

• A
• B
• C
• D

Q7:

Find the value of .

Q8:

Find the value of .

Q9:

Find the value of .

• A1
• B
• C0
• DUndefined

Q10:

Find the value of .

Q11:

Find the value of .

• A0
• B1
• C
• DUndefined

Q12:

Find the value of .

Q13:

Suppose is a point on a unit circle corresponding to the angle of . Is there another point on the unit circle representing an angle in the interval that has the same tangent value? If yes, give the angle.

• Ayes,
• Bno
• Cyes,
• Dyes,
• Eyes,

Q14:

Consider , a point on a unit circle corresponding to the angle of . Is there another point on the unit circle that has the same -coordinate as and represents an angle in the interval ? If yes, give the angle.

• AYes,
• BYes,
• CYes,
• DNo
• EYes,

Q15:

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

• Ano
• Byes,
• Cyes,
• Dyes,
• Eyes,

Q16:

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

• Ayes,
• Byes,
• Cno
• Dyes,
• Eyes,

Q17:

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

• A
• B
• C
• D

Q18:

Consider that is a point on a unit circle corresponding to the angle of . Is there another point on the unit circle that has the same -coordinate as and represents an angle in the interval ? If yes, give the angle.

• Ayes,
• Bno
• Cyes,
• Dyes,
• Eyes,

Q19:

The terminal side of in standard position intersects the unit circle at the point with coordinates , where . Find the value of .

• A
• B
• C
• D

Q20:

The terminal side of intersects the unit circle at the point where . Find the value of giving the answer to the nearest second and the value of as a fraction.

• A and
• B and
• C and
• D and

Q21:

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

• A
• B
• C
• D

Q22:

In the given figure, point is on the unit circle and lies in the interval .

From the right triangle , what ratio gives ?

• A
• B
• C
• D
• E

Which of the following is true concerning the triangles and ?

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

What is the scale factor of to ?

• A
• B
• C
• D
• E

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

• A
• B
• C
• D
• E

Q23:

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

Express and as functions of the measure of angle in radians.

• A
• B
• C
• D
• E

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

• A
• B
• C
• D
• E

Q24:

In the given figure, point is on the unit circle and lies in the interval .

Is positive or negative?

• Apositive
• Bnegative

Which of the following is true concerning the triangles , and ?

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

Q25:

The terminal side of angle in standard position intersects with the unit circle at the point, where . Find the value of .

• A
• B
• C
• D
• E