7.1: The Unit Circle (2024)

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    The core concepts of trigonometry are developed from a circle with radius equal to \(1\) unit, drawn in the \(xy\)-coordinate plane, centered at the origin. This circle is given a name: the unit circle (Figure \(7.1.1\) below). Just like a \(12\)-hour clock with values of time from \(1\) to \(12\), trigonometric functions are periodic, meaning the same values are reproduced with each \(360˚\) revolution.

    7.1: The Unit Circle (2)
    7.1: The Unit Circle (3)

    An angle is in standard position (see Figure \(7.1.2\) above) if its initial side is along the positive \(x\)-axis and its vertex is at the origin: point \((0,0)\). The following angles are in standard position. An angle that rotates in the counter-clockwise direction is a positive angle. An angle that rotates in the clockwise direction is a negative angle.

    7.1: The Unit Circle (4)
    7.1: The Unit Circle (5)
    7.1: The Unit Circle (6)

    Coterminal Angles

    Two or more standard angles that share common terminal sides are said to be coterminal angles. For example, \(30˚\) and \(390˚\) are coterminal angles.

    7.1: The Unit Circle (7)

    More formally: Every angle \(B\) is coterminal with angle \(A\) where \(B = A + 360˚k\), \(k =\) any integer.

    Example 7.1.1

    The expression \(315˚ + 360˚k\) gives the angles coterminal with \(315˚\). State the coterminal angles the equation generates using the following \(k\)-values: \(k = −2, −1, 1, 2\). Then sketch the angles.

    Solution

    We substitute the given \(k\)-values into the expression \(315˚ + 360˚k\).

    \(k = -2\) \(k = -1\) \(k = 1\) \(k = 2\)
    \(315˚ + 360˚(\textcolor{red}{-2}) = -405˚\) \(315˚ + 360˚(\textcolor{red}{-1}) = -45˚\) \(315˚ + 360˚(\textcolor{red}{1}) = -675˚\) \(315˚ + 360˚(\textcolor{red}{2}) = 1035˚\)

    Below, the angles are sketched. Do you see that each angle shares the same terminal side? All four angles are coterminal with \(315˚\), and coterminal with each other.

    7.1: The Unit Circle (8)

    Circle Centered at the Origin

    Every ordered pair \((x, y)\) on a circle is associated with a right triangle. The right triangle has horizontal distance \(x\), vertical distance \(y\), and hypotenuse = radius = \(r\).

    7.1: The Unit Circle (9)

    Equation of a Circle

    The equation of a circle of radius \(r\) centered at the origin:

    \[x^2 + y^2 = r^2\]

    Note: the \(x\)-coordinate and the \(y\)-coordinate can take on negative values, depending on the quadrant of the terminal side of the angle.

    Example 7.1.2

    Find the \(y\)-coordinate of point A \(\left(−\dfrac{5}{9} , y \right)\) if point A lies in QIII on the unit circle.

    Solution

    The unit circle has radius \(r = 1\). Trigonometry weds algebra and geometry with visual sketches. Create a sketch before jumping into a solution. This helps you see the answer.

    7.1: The Unit Circle (10)

    \(\begin{array} &\left(−\dfrac{5}{9}\right)^2 + y^2 &= 1^2 &\text{Substitute \(x = −\dfrac{5}{9}\) and \(r = 1\) into equation of a circle.} \\ \dfrac{25}{81} + y^2 &= 1 &\text{Simplify.} \\ y^2 &= 1 − \dfrac{25}{81} &\text{Subtract \(\dfrac{25}{81}\) to each side.} \\ y^2 &= \dfrac{81}{81} − \dfrac{25}{81} &\text{Find the LCD.} \\ y^2 &= \dfrac{56}{81} &\text{Simplify.} \\ \sqrt{y^2} &= ±\sqrt{\dfrac{56}{81}} &\text{Square root both sides.} \\ y &= − \dfrac{\sqrt{56}}{9} &\text{Choose the correct sign value. \(y < 0\) in QIII.} \\ y &= − \dfrac{2\sqrt{14}}{9} &\text{Simplify the radical.} \end{array}\)

    Special Right Triangles

    Using the Pythagorean Theorem, one can drive the following two templates for special right triangles: \(45˚\)-\(45˚\)-\(90˚\) triangles and \(30˚\)-\(60˚\)-\(90˚\) triangles.

    7.1: The Unit Circle (11)

    Since right triangles and circles are inextricably tied to each other, the acute angles \(30˚\), \(45˚\), \(60˚\) are frequently useful for finding exact values in trigonometry; these solutions do not require the use of a calculator.

    Examples of special right triangles and their solutions, can be viewed in these videos:

    Try It! (Exercises)

    For #1-5, draw the angle \(\theta\) in standard position.

    1. \(\theta = 210˚\)
    2. \(\theta = −300˚\)
    3. \(\theta = 150˚\)
    4. \(\theta = 270˚\)
    5. \(\theta = −135˚\)

    For #6-10, state any two angles coterminal with the given angle \(\theta\). Give one positive coterminal angle and one negative coterminal angle.

    1. \(\theta = 30˚\)
    2. \(\theta = 60˚\)
    3. \(\theta = 90˚\)
    4. \(\theta = 180˚\)
    5. \(\theta = 240˚\)

    For #11-15, state the equation of the circle with the given radius.

    1. \(r = 4\)
    2. \(r = \dfrac{1}{2}\)
    3. \(r = 3 \sqrt{2}\)
    4. \(r = \dfrac{\sqrt{6}}{2}\)
    5. \(r = \sqrt{\dfrac{101}{62}}\)
    6. Fill in the blanks: A unit circle is a circle with \(\underline{\;\;\;\;\;\;\;\;\;\;}\) equal to one unit. The circle is centered at \(\underline{\;\;\;\;\;\;\;\;\;\;}\). The circle has equation: \(\underline{\;\;\;\;\;\;\;\;\;\;}\).

    For #17-24, The given point lies on a circle with given radius and terminal side in the given quadrant. Find the missing coordinate of the given ordered pair.

    Point on Circle Radius Quadrant in which \(\theta\) terminates.
    17. \((2, y)\) \(r = 5\) \(\theta ∈\) QI
    18. \((x, \sqrt{6})\) \(r = 3\) \(\theta ∈\) QII
    19. \((−3\sqrt{7}, y)\) \(r = \sqrt{77}\) \(\theta ∈\) QIII
    20. \((x, −4\sqrt{5})\) \(r = 3\sqrt{15}\) \(\theta ∈\) QIV
    21. \(\left( \dfrac{3}{4} , y \right)\) \(r=1\) \(\theta ∈\) QI
    22. \(\left(x, \dfrac{3}{16} \right)\) \(r=1\) \(\theta ∈\) QII
    23. \(\left(− \dfrac{\sqrt{5}}{4} , y \right)\) \(r=1\) \(\theta ∈\) QIII
    24. \(\left(x, − \dfrac{3\sqrt{2}}{8} \right)\) \(r=1\) \(\theta ∈\) QIV

    For #25-29, find the fraction of a full revolution. Then sketch the angle.

    1. \(\dfrac{1}{4} \cdot 360˚\)
    2. \(\dfrac{1}{2} \cdot 360˚\)
    3. \(\dfrac{3}{4} \cdot 360˚\)
    4. \(\dfrac{1}{8} \cdot 360˚\)
    5. \(\dfrac{1}{6} \cdot 360˚\)

    For #30-35, solve for the \(2\) missing side lengths. Give exact answers. No calculators.

    1. 7.1: The Unit Circle (12)
    2. 7.1: The Unit Circle (13)
    3. 7.1: The Unit Circle (14)
    4. 7.1: The Unit Circle (15)
    5. 7.1: The Unit Circle (16)
    6. 7.1: The Unit Circle (17)

    For #36-38, Find the coordinates of the ordered pair \((x, y)\) on the unit circle with the given standard angle. Use special right triangle relationships. Give exact values of \(x\) and \(y\).

    1. 7.1: The Unit Circle (18)
    2. 7.1: The Unit Circle (19)
    3. 7.1: The Unit Circle (20)
    7.1: The Unit Circle (2024)
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