Worksheet: Trigonometric Functions of Standard Position Angle

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In this worksheet, we will practice finding the values of the six trigonometric functions of an angle θ given in standard position.

Q1:

The terminal side of 𝜃 in standard position intersects with the unit circle at the point 𝐵 with coordinates 817,1517. Find sec𝜃.

  • A 1 7 8
  • B − 1 7 8
  • C 1 7 1 5
  • D 1 5 8

Q2:

The terminal side of ∠𝐴𝑂𝐵 in standard position intersects with the unit circle at the point 𝐵 with coordinates (−𝑥,−𝑥) where 𝑥 is a postive number. Find sin𝜃.

  • A − 1 √ 2
  • B1
  • C 1 √ 2
  • D − 1

Q3:

The terminal side of 𝜃 in standard position intersects with the unit circle at the point 𝐵 with coordinates 45,−35. Find cot𝜃.

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

Q4:

The terminal side of 𝜃 in standard position intersects with the unit circle at the point 𝐵 with coordinates 817,1517. Find tan𝜃.

  • A 1 5 8
  • B − 1 5 8
  • C 1 7 8
  • D 1 7 1 5

Q5:

Two points 𝐴 and 𝐵 lie on the unit circle centered at the origin 𝑂 such that the coordinate of 𝐴 are (1,0) and ∠𝐴𝑂𝐵=𝜋3.

Which of the following properties of △𝑂𝐵𝐴 do you need to use to find the coordinates of point 𝐵𝜋3,𝜋3cossin.

  • AAll three internal angles are equal.
  • BThe height of an equilateral triangle is half its median.
  • CThe height of an equilateral triangle is equal to its side.
  • DThe height of an equilateral triangle is also its median.
  • EAll three sides are equal.

Hence, find cos𝜋3 and sin𝜋3.

  • A c o s s i n 𝜋 3 = 1 2 , 𝜋 3 = − √ 3 2
  • B c o s s i n 𝜋 3 = 1 3 , 𝜋 3 = √ 3 3
  • C c o s s i n 𝜋 3 = √ 3 2 , 𝜋 3 = 1 2
  • D c o s s i n 𝜋 3 = 1 4 , 𝜋 3 = √ 3 4
  • E c o s s i n 𝜋 3 = 1 2 , 𝜋 3 = √ 3 2

The line 𝑂𝐵 intersects the line 𝑥=1 at 𝑇(1,𝑦). The triangle 𝐴𝑇𝐶 is formed by reflecting △𝑂𝑇𝐴 in the line 𝐴𝑇. What kind of triangle is the triangle 𝑂𝑇𝐶?

  • AIsosceles
  • BEquilateral
  • CRight
  • DScalene

Find 𝐴𝑇(i.e., tan𝜋3).

  • A 2 √ 3
  • B1
  • C √ 3
  • D 3 4
  • E √ 3 3

Q6:

Find the value of tan𝜃 where 𝜃 is the measure of an angle in standard position whose terminal side passes through (5,0).

Q7:

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

Q8:

Find the value of cos𝜃 where 𝜃 is the measure of an angle in standard position whose terminal side passes through (4,0).

Q9:

Find cot(180+𝜃)∘ given 𝜃 is in standard position and its terminal side passes through the point −2129,−2029.

  • A − 2 0 2 1
  • B 2 0 2 1
  • C − 2 1 2 0
  • D 2 1 2 0

Q10:

Find sec(360−𝜃)∘ given 𝜃 is in standard position and its terminal side passes through the point 35,−45.

  • A 3 5
  • B − 3 5
  • C 5 3
  • D − 5 3

Q11:

Find sin𝜋2−𝜃 given 𝜃 is in standard position and its terminal side passes through the point 45,35.

  • A 4 5
  • B − 5 4
  • C 5 4
  • D − 4 5

Q12:

The terminal side of the angle (90−𝜃)∘ intersects the unit circle at the point 35,𝑘 where 𝜃 is a positive acute angle drawn in standard position. Find the value of 𝑘.

  • A 4 5
  • B − 4 5
  • C 5 4
  • D − 5 4

Q13:

The terminal side of 𝜃 in standard position intersects with the unit circle at the point 𝐵 with coordinates 35,𝑘. Find csc𝜋2−𝜃.

  • A 3 5
  • B − 5 3
  • C − 3 5
  • D 5 3

Q14:

Find sin3𝜋2−𝜃 given 𝜃 is in standard position and its terminal side passes through the point 817,−1517.

  • A − 1 7 8
  • B − 8 1 7
  • C 8 1 7
  • D 1 7 8

Q15:

The terminal side of the angle (90−𝜃)∘ intersects the unit circle at the point 1123,𝐾 where 𝜃 is a positive acute angle drawn in standard position. Find the value of 𝜃 giving the answer to the nearest second.

  • A 2 8 3 4 ′ 1 9 ′ ′ ∘
  • B 6 1 2 5 ′ 4 1 ′ ′ ∘
  • C 1 1 8 3 4 ′ 1 9 ′ ′ ∘
  • D 1 5 1 2 5 ′ 4 1 ′ ′ ∘

Q16:

If the angle 𝜃 is in the standard position, cos𝜃=−√22, and sin𝜃=−√22, is it possible for 𝜃 to measure 135∘?

  • AYes
  • BNo

Q17:

The terminal side of 𝜃 intersects the unit circle at the point (9𝑎,12𝑎) where 𝑎>0. Find the exact value of sec(−𝜃).

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

Q18:

The terminal side of 𝜃 intersects the unit circle at the point (−20𝑥,−5𝑥) where 𝑥>0. Find the value 𝜃 giving the answer to the nearest second.

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

Q19:

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

Q20:

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

  • AUndefined
  • B1
  • C0
  • D − 1

Q21:

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

  • A − 1
  • B0
  • CUndefined
  • D1

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