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.

  • A178
  • B−178
  • C1715
  • D158

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
  • C1√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−43
  • B43
  • C−53
  • D−34

Q4:

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

  • A158
  • B−158
  • C178
  • D1715

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.

  • Acossin𝜋3=12,𝜋3=−√32
  • Bcossin𝜋3=13,𝜋3=√33
  • Ccossin𝜋3=√32,𝜋3=12
  • Dcossin𝜋3=14,𝜋3=√34
  • Ecossin𝜋3=12,𝜋3=√32

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).

  • A2√3
  • B1
  • C√3
  • D34
  • E√33

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−2021
  • B2021
  • C−2120
  • D2120

Q10:

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

  • A35
  • B−35
  • C53
  • D−53

Q11:

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

  • A45
  • B−54
  • C54
  • D−45

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.

  • A45
  • B−45
  • C54
  • D−54

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.

  • A35
  • B−53
  • C−35
  • D53

Q14:

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

  • A−178
  • B178
  • C817
  • D−817

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.

  • A2834′19′′∘
  • B6125′41′′∘
  • C11834′19′′∘
  • D15125′41′′∘

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).

  • A34
  • B35
  • C53
  • D54

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.

  • A1942′10′′∘
  • B34557′50′′∘
  • C16557′50′′∘
  • D142′10′′∘

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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