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IBMathsDPApplications & Interpretation (AI)HLFlashcards3. Geometry & TrigonometryTrigonometric Identities & Equations

Trigonometric Identities & Equations (DP IB Applications & Interpretation (AI))

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  • FrontThe Unit Circle

    What is the unit circle?

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Cards in this collection (29)

  • What is the unit circle?

    The unit circle is a circle with radius 1 and centre (0, 0), which can be used to calculate trigonometric values.

  • How can the unit circle be used to find trig values?

    Trig values can be found by creating a right-angled triangle between the x-axis and the radius of the unit circle.

    The unit circle with centre (0, 0) and radius 1. A right-angled triangle is inscribed within the circle, showing the lengths that represent the different trig functions.

  • How are angles measured on the unit circle?

    Angles on the unit circle are always measured from the positive x-axis.

    Positive angles are measured anti-clockwise and negative angles are measured clockwise.

  • What does the x-coordinate represent on the unit circle?

    The x-coordinate on the unit circle gives the value of cos space theta.

  • What does the y-coordinate represent on the unit circle?

    The y-coordinate on the unit circle gives the value of sin space theta.

  • How is tan space theta represented on the unit circle?

    On the unit circle, tan space theta is represented by the gradient of the line from the origin to the point (x, y).

  • What is the CAST diagram?

    The CAST diagram is a mnemonic for remembering which trigonometric functions are positive in each quadrant of the unit circle (Cosine, All, Sine, Tangent).

    CAST diagram. The unit circle with each consecutive quadrant, (starting with the bottom right quadrant and going anti-clockwise), labelled C, A, S and T.

  • True or False?

    Trigonometric functions have only one input for each output.

    False.

    Trigonometric functions can have multiple inputs for each output.

  • What is a primary value in a trigonometric equation?

    The primary value is the first solution given by a calculator for a trigonometric equation.

  • What is the first step in finding secondary values for a trigonometric equation using the unit circle?

    The first step in finding secondary values for a trigonometric equation using the unit circle, is to draw the angle into the first quadrant using the x or y coordinate to help you.

  • True or False?

    The unit circle can be used to calculate trigonometric values for angles greater than 90°.

    True.

    The unit circle can be used to calculate trigonometric values for angles greater than 90°.

  • True or False?

    The unit circle can be used to construct the sine curve.

    True.

    The unit circle can be used to construct the sine curve by plotting the y-coordinate from the unit circle as the x-coordinate on the graph y equals sin x.

    A diagram showing how the unit circle generates the sine wave. Key points are labeled at 0, π/2, π, 3π/2, and 2π on the x-axis, with corresponding sine values from -1 to 1 on the y-axis.

  • What is a trigonometric identity?

    A trigonometric identity is a statement that is true for all values of θ or x in trigonometry.

  • State the identity for tan space theta.

    The tan identity is tan space theta equals fraction numerator sin space theta over denominator cos space theta end fraction.

    This is given in your exam formula booklet.

  • State the Pythagorean identity.

    The Pythagorean identity is sin squared theta plus cos squared theta equals 1.

    This is given in your exam formula booklet.

  • True or False?

    Trigonometric identities can be used to prove double angle formulae.

    True.

    Trigonometric identities can be used to prove further identities such as the double angle formulae.

  • Which function is this the graph of?

    Periodic graph ranging from -180 to 360 on the x-axis. Vertical asymptotes at -90, 90, 270 degrees. The curve repeats every 180 degrees, transitioning from negative to positive infinity.

    The graph is of the tangent function.

    It is a periodic graph with the following features:

    • x-intercepts at -180º, 0º, 180º, 360º (and every 180º in either direction),

    • a y-intercept at (0, 0),

    • asymptotes at -90º, 90º, 270º (and every 180º in either direction),

    • and a range of y element of straight real numbers.

  • Which function is this the graph of?

    Periodic graph ranging from -180 to 360 on the x-axis. The curve intersects the x-axis at 90 and every 180 from there in either direction. The graph intercepts the y-axis at (0, 1) and cycles between a maximum of 1 and a minimum of -1.

    The graph is of the cosine function.

    It is a periodic graph with the following features:

    • x-intercepts at -90º, 90º, 270º (and every 180º in either direction),

    • a y-intercept at (0, 1),

    • and a range of -1 ≤ y ≤ 1.

  • Which function is this the graph of?

    Periodic graph ranging from -180 to 360 on the x-axis. The curve intersects the x-axis at 0 and every 180 from there in either direction. The graph intercepts the y-axis at (0, 0) and cycles between a maximum of 1 and a minimum of -1.

    The graph is of the sine function.

    It is a periodic graph with the following features:

    • x-intercepts at -180º, 0º, 180º, 360º (and every 180º in either direction),

    • a y-intercept at (0, 0),

    • and a range of -1 ≤ y ≤ 1.

  • True or False?

    The graph of sin x passes through the origin.

    True.

    The graph of sin x passes through the origin.

  • What is the period of tan x?

    The period of tan x is 180º (or pi radians).

  • What is the period of sin x and cos x?

    The period of both sin x and cos x is 360º (or 2pi radians).

  • Define the range of sin x and cos x.

    The range of both sin x and cos x is -1 ≤ y ≤ 1.

  • Define the range of tan x.

    The range of tan x is y element of straight real numbers. I.e., tan x can take any real number value (positive, negative or zero).

  • What are the equations of the asymptotes in the graph of tan x, negative pi less or equal than x less or equal than 2 pi.

    The equations of the asymptotes in the graph of tan x, negative pi less or equal than x less or equal than 2 pi are

    x equals negative pi over 2, x equals pi over 2 and x equals fraction numerator 3 pi over denominator 2 end fraction.

  • What is the relationship between sin(-x) and sin(x)?

    The relationship between sin(-x) and sin(x) is sin(-x) = -sin(x).

  • What is the relationship between cos(-x) and cos(x)?

    The relationship between cos(-x) and cos(x) is cos(-x) = cos(x).

  • True or False?

    tan(x) = tan(x ± 180°)

    True.

    tan(x) = tan(x ± 180°) or tan(x ± pi)

  • What is the relationship between sin(x) and sin(pi - x)?

    The relationship between sin(x) and sin(pi - x), is sin(x) = sin(pi - x), (or sin(x) = sin(180° - x) ).

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