11th Class Trigonometric Functions
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Questions and Answers

What is the amplitude of a periodic function?

  • The height from the centre to the maximum or minimum values (correct)
  • The horizontal shift of the function
  • The distance between peaks and troughs
  • The length of one cycle on the x-axis

    • Sin is positive in the 2nd quadrant according to the ASTC rule.

    True

    What is the angle that gives θ in the triangle in the 3rd quadrant?

    180° + θ

    ASTC rule stands for 'All Stations To ________'.

    <p>Central</p> Signup and view all the answers

    Match the trigonometric functions with their inverse functions:

    <p>sin = sin⁻¹ cos = cos⁻¹ tan = tan⁻¹</p> Signup and view all the answers

    What is the value of $a$ in the Pythagorean theorem equation $a^2 + 4 = 49$?

    <p>5</p> Signup and view all the answers

    Which trigonometric ratio is equivalent to $\cot \theta$?

    <p>$\tan \theta$</p> Signup and view all the answers

    For any value of $\theta$, $\cos^2 \theta + \sin^2 \theta = ___$

    <p>1</p> Signup and view all the answers

    Match the following Pythagorean identities:

    <p>$\cos^2 \theta + \sin^2 \theta$ = 1 $1 + \tan^2 \theta$ = $\sec^2 \theta$ $1 + \cot^2 \theta$ = $\csc^2 \theta$</p> Signup and view all the answers

    What is the value of $m$ in the equation $\sec 55^\circ = \csc (2m - 15)^\circ$?

    <p>25</p> Signup and view all the answers

    Find all quadrants where cos θ > 0, tan θ > 0, and sin θ > 0. (Select all that apply)

    <p>1st quadrant</p> Signup and view all the answers

    Which quadrant is the angle 240° in?

    <p>3rd quadrant</p> Signup and view all the answers

    Find the exact value of cos 240°.

    <p>-1/2</p> Signup and view all the answers

    Which quadrant is the angle 315° in?

    <p>4th quadrant</p> Signup and view all the answers

    Find the exact value of sin 315°.

    <p>1/√2</p> Signup and view all the answers

    Which quadrant is the angle 120° in?

    <p>2nd quadrant</p> Signup and view all the answers

    Find the exact value of tan 120°.

    <p>√3</p> Signup and view all the answers

    Which quadrant is the angle -225° in?

    <p>3rd quadrant</p> Signup and view all the answers

    Find the exact value of sin (-225°).

    <p>-1/√2</p> Signup and view all the answers

    Which quadrant is the angle -330° in?

    <p>4th quadrant</p> Signup and view all the answers

    Find the exact value of cos (-330°).

    <p>-√3/2</p> Signup and view all the answers

    Find the exact value of tan (-120°).

    <p>√3</p> Signup and view all the answers

    Write cos 2x sin 5x as a sum of trigonometric ratios.

    <p>sin 7x + sin 3x</p> Signup and view all the answers

    Find the exact value of sin 75° sin 15°.

    <p>1/4</p> Signup and view all the answers

    Find the exact value of 2tan15° / (1 + tan²15°).

    <p>1</p> Signup and view all the answers

    Prove that cot(A/2) - 2cotA = tan(A/2).

    <p>tan(A/2)</p> Signup and view all the answers

    Find the exact value of sin 30°.

    <p>1/2</p> Signup and view all the answers

    Find an expression for sin (x + y) + sin (x − y).

    <p>2sin(x)cos(y)</p> Signup and view all the answers

    Write an expression for cos (x + y) − cos (x − y).

    <p>2sin(x)sin(y)</p> Signup and view all the answers

    In which quadrant is the angle 3π/4 located?

    <p>2nd quadrant</p> Signup and view all the answers

    Find the exact value of cos(3π/4).

    <p>-√2/2</p> Signup and view all the answers

    Simplify: 2cos(3x)sin(3x).

    <p>sin(6x)</p> Signup and view all the answers

    Show that $\frac{5\pi}{6} = \frac{\pi}{6}$.

    <p>$5\pi = \pi + 6$</p> Signup and view all the answers

    In which quadrant is the angle $\frac{5\pi}{6}$?

    <p>2nd quadrant</p> Signup and view all the answers

    Find the exact value of $\sin \frac{5\pi}{6}$.

    <p>-$\frac{1}{2}$</p> Signup and view all the answers

    Show that $\frac{7\pi}{4} = 2\pi - \frac{\pi}{4}$.

    <p>$7\pi = 8\pi - 4$</p> Signup and view all the answers

    In which quadrant is the angle $\frac{7\pi}{4}$?

    <p>1st quadrant</p> Signup and view all the answers

    Find the exact value of $\tan \frac{7\pi}{4}$.

    <p>1</p> Signup and view all the answers

    What is the domain of the function y = tan(x)?

    <p>(-∞, ∞) except for odd multiples of 90° (90°, 270°, 450°, ...)</p> Signup and view all the answers

    What is the range of the function y = tan(x)?

    <p>(-∞, ∞)</p> Signup and view all the answers

    What is the domain of the function y = cosec(x)?

    <p>(-∞, ∞) except for 0°, 180°, 360°, ...</p> Signup and view all the answers

    What is the range of the function y = cosec(x)?

    <p>(-∞, ∞)</p> Signup and view all the answers

    What are the vertical asymptotes for the function y = sec(x)?

    <p>90°, 270°, 450°, ...</p> Signup and view all the answers

    The function y = cot(x) will have a vertical asymptote at 90°.

    <p>True</p> Signup and view all the answers

    Study Notes

    Trigonometric Functions

    • Trigonometric functions are used to model real-life situations, such as tides, annual temperatures, and phases of the Moon.
    • These functions are periodic, meaning they repeat regularly.

    Angles of Any Magnitude

    • Angles can be measured in any size, not just acute and obtuse angles.
    • Angles can be measured in different quadrants:
      • 1st quadrant: 0° to 90°, all ratios are positive
      • 2nd quadrant: 90° to 180°, sin is positive, cos and tan are negative
      • 3rd quadrant: 180° to 270°, tan is positive, sin and cos are negative
      • 4th quadrant: 270° to 360°, cos is positive, sin and tan are negative
    • The ASTC rule (All Stations To Central) helps to remember the signs of the ratios in each quadrant.

    Trigonometric Ratios

    • Trigonometric ratios can be found for angles greater than 360° by turning around the circle more than once.
    • Negative angles can be measured in the opposite direction from positive angles.
    • The ASTC rule also works for negative angles.

    Trigonometric Identities

    • Reciprocal trigonometric ratios are the reciprocals of the sine, cosine, and tangent ratios.
    • Examples of reciprocal trigonometric ratios:
      • sin-1, cos-1, and tan-1
      • cosecant, secant, and cotangent (which are the reciprocals of sine, cosine, and tangent respectively)

    Terminology

    • Amplitude: the height from the center of a periodic function to the maximum or minimum values.
    • Centre: the mean value of a periodic function that is equidistant from the maximum and minimum values.
    • Period: the length of one cycle of a periodic function.
    • Phase: a horizontal shift (translation).
    • Identity: an equation that shows the equivalence of two algebraic expressions for all values of the variables.
    • Inverse trigonometric functions: the sin-1, cos-1, and tan-1 functions, which are the inverse functions of the sine, cosine, and tangent functions respectively.### Reciprocal Trigonometric Ratios
    • The reciprocal trigonometric ratios are: cosecant (cosec), secant (sec), and cotangent (cot)
    • The formulas for these ratios are:
      • cosec θ = 1 / sin θ
      • sec θ = 1 / cos θ
      • cot θ = 1 / tan θ
    • These ratios have the same signs as their related ratios in the different quadrants

    Complementary Angles

    • If ∠B = θ, then ∠A = 90° - θ
    • Angles ∠B and ∠A are complementary because they add up to 90°
    • The trigonometric ratios of complementary angles are equal:
      • sin θ = cos (90° - θ)
      • cos θ = sin (90° - θ)
      • tan θ = cot (90° - θ)
      • sec θ = cosec (90° - θ)
      • cot θ = tan (90° - θ)

    Trigonometric Identities

    • The tangent identity: tan θ = sin θ / cos θ
    • The Pythagorean identities:
      • cos² θ + sin² θ = 1
      • 1 + tan² θ = sec² θ
      • 1 + cot² θ = cosec² θ
    • These identities can be rearranged to give:
      • cos² θ = 1 - sin² θ
      • sin² θ = 1 - cos² θ

    Sums and Differences of Angles

    • The formulas for the trigonometric ratios of sums and differences of angles are:
      • cos (A - B) = cos A cos B + sin A sin B
      • cos (A + B) = cos A cos B - sin A sin B
      • sin (A + B) = sin A cos B + cos A sin B
      • sin (A - B) = sin A cos B - cos A sin B### Sum and Difference Identities
    • The sum and difference identities for trigonometric functions are:
      • sin(A + B) = sin A cos B + cos A sin B
      • sin(A - B) = sin A cos B - cos A sin B
      • cos(A + B) = cos A cos B - sin A sin B
      • cos(A - B) = cos A cos B + sin A sin B
      • tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
      • tan(A - B) = (tan A - tan B) / (1 + tan A tan B)

    Double Angle Identities

    • The double angle identities for trigonometric functions are:
      • sin 2A = 2 sin A cos A
      • cos 2A = cos^2 A - sin^2 A
      • tan 2A = 2 tan A / (1 - tan^2 A)

    Product to Sum Identities

    • The product to sum identities for trigonometric functions are:
      • cos A cos B = [cos(A + B) + cos(A - B)] / 2
      • sin A sin B = [cos(A - B) - cos(A + B)] / 2
      • sin A cos B = [sin(A + B) + sin(A - B)] / 2
      • cos A sin B = [sin(A + B) - sin(A - B)] / 2

    The t-Formulas

    • The t-formulas express sin A, cos A, and tan A in terms of t = tan(A/2):
      • sin A = 2t / (1 + t^2)
      • cos A = (1 - t^2) / (1 + t^2)
      • tan A = 2t / (1 - t^2)

    Examples and Exercises

    • Various examples and exercises are provided to practice and apply the trigonometric identities.

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