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Worksheet: Directed Angles In Standard Position

Q1:

The terminal side of angle πœƒ in standard position intersects with the unit circle at the point, ( 2 π‘Ž , 3 π‘Ž ) where 0 < πœƒ < πœ‹ 2 . Find the value of π‘Ž .

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

Q2:

The terminal side of πœƒ intersects the unit circle at the point ( βˆ’ 2 0 π‘₯ , βˆ’ 5 π‘₯ ) where π‘₯ > 0 . Find the value πœƒ giving the answer to the nearest second.

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

Q3:

Is the angle in standard position?

  • Ayes
  • Bno

Q4:

Is the angle in standard position?

  • Ano
  • Byes

Q5:

Is the angle in standard position?

  • Ano
  • Byes

Q6:

Is the angle in standard position?

  • Ano
  • Byes

Q7:

Is the angle in standard position?

  • Ayes
  • Bno

Q8:

Is the angle in standard position?

  • Ayes
  • Bno

Q9:

Is the angle in standard position?

  • Ano
  • Byes

Q10:

Find the value of c o t πœƒ where πœƒ is the measure of an angle in standard position whose terminal side passes through ( 0 , βˆ’ 6 ) .

Q11:

The terminal side of ∠ 𝐴 𝑂 𝐡 in standard position intersects the unit circle 𝑂 at the point 𝐡 with coordinates ο€Ώ 3 √ 1 0 , 𝑦  , where 𝑦 > 0 . Find the value of s i n 𝐴 𝑂 𝐡 .

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

Q12:

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

Q13:

Does the Coordinate express an angle in standard position?

  • Ano
  • Byes

Q14:

The terminal side of πœƒ in standard position intersects with the unit circle at the point 𝐡 with coordinates ο€Ό βˆ’ 4 5 , βˆ’ 3 5  . Find the value of πœƒ giving the answer to the nearest second.

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

Q15:

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

  • A1
  • B0
  • C βˆ’ 1
  • DUndefined

Q16:

In the given figure, point 𝑀 is on the unit circle and πœƒ lies in the interval ο€» 0 , πœ‹ 2  .

From the right triangle 𝑂 𝑀 𝑋 , what ratio gives t a n πœƒ ?

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

Which of the following is true concerning the triangles 𝑂 𝑀 𝑋 and 𝑂 𝑇 𝐴 ?

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

What is the scale factor of β–³ 𝑂 𝑀 𝑋 to β–³ 𝑂 𝑇 𝐴 ?

  • A s i n πœƒ
  • B 1 πœƒ s i n
  • C 1 πœƒ c o s
  • D 1 πœƒ t a n
  • E c o s πœƒ

Use your answers to the previous parts to determine the length of 𝐴 𝑇 in terms of s i n πœƒ and c o s πœƒ .

  • A s i n c o s πœƒ πœƒ
  • B 1 πœƒ t a n
  • C c o s πœƒ
  • D c o s s i n πœƒ πœƒ
  • E 1 πœƒ c o s

Q17:

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

Express π‘Ž and 𝑏 as functions of the measure of angle πœƒ in radians.

  • A π‘Ž = πœƒ , 𝑏 = πœƒ t a n s i n
  • B π‘Ž = πœƒ , 𝑏 = πœƒ s i n c o s
  • C π‘Ž = πœƒ , 𝑏 = πœƒ c o s t a n
  • D π‘Ž = πœƒ , 𝑏 = πœƒ c o s s i n
  • E π‘Ž = βˆ’ πœƒ , 𝑏 = βˆ’ πœƒ c o s s i n

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

  • A 8 πœ‹ 3
  • B 3 πœ‹ 5
  • C 3 πœ‹ 8
  • D 5 πœ‹ 3
  • E 4 πœ‹ 3

Q18:

In the given figure, point 𝑀 is on the unit circle and πœƒ lies in the interval ο€» πœ‹ 2 , πœ‹  .

Is t a n πœƒ positive or negative?

  • Anegative
  • Bpositive

Which of the following is true concerning the triangles 𝑂 𝑋 𝑀 , 𝑂 𝐴 𝑇 , and 𝑂 𝐢 𝐴 ?

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

Q19:

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

  • A1
  • B0
  • C βˆ’ 1
  • DUndefined