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Unit circle and inverse values

Use signed unit-circle coordinates and exact ratios, then select principal inverse values without losing the original angle context.

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Enlarged diagram

Coordinates define sine and cosine for any angle

A right-triangle ratio starts with an acute angle. The unit circle extends that idea to any angle: begin on the positive horizontal axis, turn anticlockwise for a positive angle and clockwise for a negative angle. The point reached on the circle of radius 1 has coordinates (cos θ, sin θ).

Cosine is the horizontal coordinate and sine the vertical coordinate. In quadrant II the point is left of the vertical axis and above the horizontal axis, so cosine is negative and sine positive. At 150° the reference angle is 30°, giving

cos 150° = -(square root of 3)/2, sin 150° = 1/2.

A unit-circle point can have negative cosine and positive sine

A circle of radius one centred at the origin, with equal physical scales on both axes. A radius ends in quadrant II at approximately (−0.866, 0.5); dotted lines project that point onto each coordinate axis.

Scroll across the figure to read all labels.

A circle of radius one centred at the origin, with equal physical scales on both axes. A radius ends in quadrant II at approximately (−0.866, 0.5); dotted lines project that point onto each coordinate axis.A circle of radius one centred at the origin, with equal physical scales on both axes. A radius ends in quadrant II at approximately (−0.866, 0.5); dotted lines project that point onto each coordinate axis.
The diamond is the point reached at 150° anticlockwise from the positive horizontal axis: (cos 150°, sin 150°) = (−√3/2, 1/2). The dashed radius has length 1. Dotted projections show the negative horizontal coordinate and positive vertical coordinate.
Open full-size graphOpen full-size graph
What to noticeCosine is the horizontal coordinate; sine is the vertical coordinate. The curve and table use decimal samples for display; the exact point coordinates follow from the 30° reference triangle. The graph does not draw the directed angle arc: 150° is measured anticlockwise from the positive horizontal axis.
View figure data
Figure data: A unit-circle point can have negative cosine and positive sine
SeriesHorizontal coordinate cos θ (dimensionless)Vertical coordinate sin θ (dimensionless)
Unit circle1.000000.00000
Unit circle0.998630.05234
Unit circle0.994520.10453
Unit circle0.987690.15643
Unit circle0.978150.20791
Unit circle0.965930.25882
Unit circle0.951060.30902
Unit circle0.933580.35837
Unit circle0.913550.40674
Unit circle0.891010.45399
Unit circle0.866030.50000
Unit circle0.838670.54464
Unit circle0.809020.58779
Unit circle0.777150.62932
Unit circle0.743140.66913
Unit circle0.707110.70711
Unit circle0.669130.74314
Unit circle0.629320.77715
Unit circle0.587790.80902
Unit circle0.544640.83867
Unit circle0.500000.86603
Unit circle0.453990.89101
Unit circle0.406740.91355
Unit circle0.358370.93358
Unit circle0.309020.95106
Unit circle0.258820.96593
Unit circle0.207910.97815
Unit circle0.156430.98769
Unit circle0.104530.99452
Unit circle0.052340.99863
Unit circle0.000001.00000
Unit circle-0.052340.99863
Unit circle-0.104530.99452
Unit circle-0.156430.98769
Unit circle-0.207910.97815
Unit circle-0.258820.96593
Unit circle-0.309020.95106
Unit circle-0.358370.93358
Unit circle-0.406740.91355
Unit circle-0.453990.89101
Unit circle-0.500000.86603
Unit circle-0.544640.83867
Unit circle-0.587790.80902
Unit circle-0.629320.77715
Unit circle-0.669130.74314
Unit circle-0.707110.70711
Unit circle-0.743140.66913
Unit circle-0.777150.62932
Unit circle-0.809020.58779
Unit circle-0.838670.54464
Unit circle-0.866030.50000
Unit circle-0.891010.45399
Unit circle-0.913550.40674
Unit circle-0.933580.35837
Unit circle-0.951060.30902
Unit circle-0.965930.25882
Unit circle-0.978150.20791
Unit circle-0.987690.15643
Unit circle-0.994520.10453
Unit circle-0.998630.05234
Unit circle-1.000000.00000
Unit circle-0.99863-0.05234
Unit circle-0.99452-0.10453
Unit circle-0.98769-0.15643
Unit circle-0.97815-0.20791
Unit circle-0.96593-0.25882
Unit circle-0.95106-0.30902
Unit circle-0.93358-0.35837
Unit circle-0.91355-0.40674
Unit circle-0.89101-0.45399
Unit circle-0.86603-0.50000
Unit circle-0.83867-0.54464
Unit circle-0.80902-0.58779
Unit circle-0.77715-0.62932
Unit circle-0.74314-0.66913
Unit circle-0.70711-0.70711
Unit circle-0.66913-0.74314
Unit circle-0.62932-0.77715
Unit circle-0.58779-0.80902
Unit circle-0.54464-0.83867
Unit circle-0.50000-0.86603
Unit circle-0.45399-0.89101
Unit circle-0.40674-0.91355
Unit circle-0.35837-0.93358
Unit circle-0.30902-0.95106
Unit circle-0.25882-0.96593
Unit circle-0.20791-0.97815
Unit circle-0.15643-0.98769
Unit circle-0.10453-0.99452
Unit circle-0.05234-0.99863
Unit circle0.00000-1.00000
Unit circle0.05234-0.99863
Unit circle0.10453-0.99452
Unit circle0.15643-0.98769
Unit circle0.20791-0.97815
Unit circle0.25882-0.96593
Unit circle0.30902-0.95106
Unit circle0.35837-0.93358
Unit circle0.40674-0.91355
Unit circle0.45399-0.89101
Unit circle0.50000-0.86603
Unit circle0.54464-0.83867
Unit circle0.58779-0.80902
Unit circle0.62932-0.77715
Unit circle0.66913-0.74314
Unit circle0.70711-0.70711
Unit circle0.74314-0.66913
Unit circle0.77715-0.62932
Unit circle0.80902-0.58779
Unit circle0.83867-0.54464
Unit circle0.86603-0.50000
Unit circle0.89101-0.45399
Unit circle0.91355-0.40674
Unit circle0.93358-0.35837
Unit circle0.95106-0.30902
Unit circle0.96593-0.25882
Unit circle0.97815-0.20791
Unit circle0.98769-0.15643
Unit circle0.99452-0.10453
Unit circle0.99863-0.05234
Unit circle1.000000.00000
Radius to the 150° point0.000000.00000
Radius to the 150° point-0.866030.50000
Coordinate projections-0.866030.00000
Coordinate projections-0.866030.50000
Coordinate projections0.000000.50000
Point at 150°-0.866030.50000

The radius is still positive; the signs belong to coordinates, rather than to negative side lengths. The same point is reached at −210° or 510°, since removing or adding a complete turn leaves its coordinates unchanged.

Radians describe the same turns

For a positive angle, its radian measure is arc length divided by radius. A full circumference is 2π times the radius, so one full turn is 2π radians, or 360°. The negative sign still marks clockwise direction. Convert degrees to radians by multiplying by π/180; convert radians to degrees by multiplying by 180/π.

Thus 150° = 5π/6 radians. The values sin 150° and sin(5π/6) are the same. Entering the number π/6 on a calculator set to degrees instead evaluates an angle of about 0.524°, giving about 0.00914 rather than 1/2. State the angle unit and set the calculator accordingly.

Derive the special-angle values

In a right isosceles triangle with legs 1 and 1, the hypotenuse is √2. Dividing each coordinate by √2 gives sine and cosine 1/√2 at 45°.

Bisect an equilateral triangle of side 2. The resulting right triangle has hypotenuse 2, short leg 1 and long leg √3 by Pythagoras. At 30°, sine is 1/2 and cosine √3/2; at 60° those two values exchange roles. Tangent is sine divided by cosine.

Anglesinecosinetangent
30° or π/61/2√3/21/√3
45° or π/41/√21/√21
60° or π/3√3/21/2√3

For another quadrant, keep the reference-angle magnitudes and obtain the signs from the coordinates. At 150°, tangent is (1/2)/(- square root of 3/2) = -1/square root of 3.

Reflection in the horizontal axis changes θ to −θ, keeping the horizontal coordinate and reversing the vertical one. Hence cos(−θ) = cos θ and sin(−θ) = −sin θ. A half-turn reverses both coordinates, while a full turn leaves both unchanged. These relationships explain repeated values and graph symmetries rather than requiring a separate sign rule for each angle.

Reciprocal functions retain their own domains

The six functions use the same two coordinates, but division introduces restrictions:

FunctionCoordinate definitionRequired condition
sinevertical coordinateevery real angle
cosinehorizontal coordinateevery real angle
tangentsin θ / cos θcos θ ≠ 0
secant1 / cos θcos θ ≠ 0
cosecant1 / sin θsin θ ≠ 0
cotangentcos θ / sin θsin θ ≠ 0

At 90°, sine is 1, cosine 0, cosecant 1 and cotangent 0; tangent and secant are undefined. Defining cotangent as cos θ / sin θ establishes its value there. Writing it as 1/tan θ works only where that expression is defined, so it cannot establish cot 90° by dividing by an undefined tangent.

Check your understanding 1 Use the coordinates before taking reciprocals

Find the exact values of all six trigonometric functions at 330°. Explain the signs of sine and cosine from the point’s quadrant, and check whether any function is undefined.

Show answer

The reference angle is 30° and the point is in quadrant IV: sine is −1/2 and cosine √3/2. Thus tangent is −1/√3, secant 2/√3, cosecant −2 and cotangent −√3. Neither coordinate is zero, so all six are defined. Equivalent rationalised forms, such as −√3/3 for tangent and 2√3/3 for secant, have the same exact values.

Inverse functions choose a principal range

Many angles share one sine, cosine or tangent value, so an inverse function needs a chosen output range. A calculator returns the angle in that principal range, rather than every possible angle.

Inverse functionAllowed real inputPrincipal output in radiansPrincipal output in degrees
inverse sine−1 to 1, inclusive[−π/2, π/2][−90°, 90°]
inverse cosine−1 to 1, inclusive[0, π][0°, 180°]
inverse tangentevery real number(−π/2, π/2)(−90°, 90°)

For example, sin⁻¹ (1/2) = 30°, while cosec 30° = 2. The superscript −1 in the inverse-function notation does not mean reciprocal. No real inverse-sine value exists for input 2, because a unit-circle vertical coordinate cannot be 2.

Taking an inverse after a trigonometric function need not return the starting angle:

  • sin⁻¹(sin 150°) = 30°, because both angles have sine 1/2 and only 30° lies in the principal sine range.
  • cos⁻¹(cos 240°) = 120°, because both have cosine −1/2 and only 120° lies in the principal cosine range.
  • tan⁻¹(tan 135°) = −45°, because both have tangent −1 and only −45° lies in the principal tangent range.

The inverse returns the original angle when that angle is already in the appropriate principal range. Otherwise use the unit-circle coordinates and the requested interval to decide which angle is needed.

Check your understanding 2 Principal value or interval solution?

Find the principal value of cos⁻¹ (-1/2) in degrees, then list all solutions of cos x = -1/2 for 0° ≤ x < 360°. Explain why the two requests need different answers.

Show answer

The principal inverse-cosine value is 120°. The interval solutions are 120° and 240°. Cosine is negative in quadrants II and III; the principal range [0°,180°] selects only the first of these. The interval question includes both qualifying angles.

Check the angle unit

sin(π/6) = 1/2 uses radians. A degree symbol or a stated radian convention is part of the meaning of an angle, not a display preference. Keep exact angle multiples of π when requested, and round only a final numerical angle after choosing its quadrant and principal range.

Try it yourself 1

Recover the point and interpret the inverse values

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Problem

An angle θ satisfies cos θ = −3/5 and 180° < θ < 270°. Find its unit-circle point and the exact values of all six trigonometric functions. Then give θ, sin⁻¹(sin θ), cos⁻¹(cos θ) and tan⁻¹(tan θ) in degrees to two decimal places, explaining each range choice.

Write your answer, then compare it with the solution

Hints

Hint 1

Separate a coordinate sign from an inverse function's output range

Use the radius and the given horizontal coordinate to find the vertical magnitude, then apply the stated quadrant. Keep each inverse function’s principal range beside its numerical value.

Show full solution

  1. Recover the unit-circle point

    Method

    Use the radius 1 and horizontal coordinate −3/5.

    Reason

    The vertical coordinate’s magnitude follows from Pythagoras, while the quadrant determines its sign.

    Working

    Answer

    The vertical coordinate has square 1 − 9/25 = 16/25. In quadrant III it is negative, so sin θ = −4/5 and the point is (−3/5, −4/5).

  2. Form the six exact functions

    Method

    Use the two coordinates and their defined ratios.

    Reason

    Neither coordinate is zero, so every required denominator is non-zero.

    Working

    Answer

    Sine is −4/5, cosine −3/5, tangent 4/3, secant −5/3, cosecant −5/4 and cotangent 3/4. Tangent is positive because both coordinates are negative.

  3. Choose each requested angle in its own range

    Method

    Use a reference angle and the three principal ranges.

    Reason

    A principal inverse value need not lie in the original angle’s quadrant.

    Working

    Answer

    The acute reference angle is tan⁻¹(4/3) ≈ 53.13°, so θ ≈ 233.13°. The principal inverse-sine value is −53.13°, the inverse-cosine value 126.87° and the inverse-tangent value 53.13°. Each has the required function value and lies in its own principal range. Use unrounded calculator values before reporting two decimal places.

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