Worksheet: Directed Angles In Standard Position

In this worksheet, we will practice identifying whether the directed angle is in standard position.

Q1:

Does the ordered pair express an angle in standard position?

  • Ano
  • Byes

Q2:

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

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

Q3:

Is the angle in standard position?

  • Ayes
  • Bno

Q4:

Is the angle in standard position?

  • Ayes
  • Bno

Q5:

The terminal side of 𝐴 𝑂 𝐵 in standard position intersects the unit circle 𝑂 at the point 𝐵 with coordinates 3 1 0 , 𝑦 , where 𝑦 > 0 . Find the value of s i n 𝐴 𝑂 𝐵 .

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

Q6:

The terminal side of 𝜃 in standard position intersects with the unit circle at the point 𝐵 with coordinates 4 5 , 3 5 . Find the value of 𝜃 giving the answer to the nearest second.

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

Q7:

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

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

Q8:

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

If the angle 𝜃 at a certain time is 5 𝜋 3 , what will it be after the blade has completed half a rotation?

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

Q9:

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

Is positive or negative?

  • Anegative
  • Bpositive

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

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

Q10:

Find the value of c o t 𝜃 where 𝜃 is the measure of an angle in standard position whose terminal side passes through ( 0 , 6 ) .

Q11:

Find the value of s e c 𝜃 where 𝜃 is the measure of an angle in standard position whose terminal side passes through ( 0 , 7 ) .

  • A1
  • B0
  • C 1
  • DUndefined

Q12:

Find the value of c s c 𝜃 where 𝜃 is the measure of an angle in standard position whose terminal side passes through ( 2 , 0 ) .

  • A1
  • B0
  • C 1
  • DUndefined

Q13:

The terminal side of 𝜃 intersects the unit circle at the point ( 2 0 𝑥 , 5 𝑥 ) where 𝑥 > 0 . Find the value 𝜃 giving the answer to the nearest second.

  • A 1 4 2 1 0
  • B 1 6 5 5 7 5 0
  • C 3 4 5 5 7 5 0
  • D 1 9 4 2 1 0

Q14:

The terminal side of 𝜃 intersects the unit circle at the point 8 1 7 , 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 = 6 1 5 5 3 9 and 𝑦 = 1 5 1 7
  • B 𝑚 𝜃 = 2 4 1 5 5 3 9 and 𝑦 = 1 5 1 7
  • C 𝑚 𝜃 = 1 2 8 5 5 5 6 and 𝑦 = 8 1 5
  • D 𝑚 𝜃 = 6 1 5 5 3 9 and 𝑦 = 1 5 1 7

Q15:

Is the angle in standard position?

  • Ano
  • Byes

Q16:

Is the angle in standard position?

  • Ano
  • Byes

Q17:

Is the angle in standard position?

  • Ano
  • Byes

Q18:

Is the angle in standard position?

  • Ayes
  • Bno

Q19:

Is the angle in standard position?

  • Ano
  • Byes

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