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Questions and Answers
What is the reference angle for 215°?
What is the reference angle for 215°?
- 145°
- 35°
- 45°
- 105° (correct)
What is the value of csc A if a = 5 and b = 2√6 in triangle ABC?
What is the value of csc A if a = 5 and b = 2√6 in triangle ABC?
- √3
- 2.5
- 2√3 (correct)
- 5/2√3
What is the exact value of csc 780°?
What is the exact value of csc 780°?
- 1
- 0.5
- Undefined (correct)
- √2
What is the value of tan 75°?
What is the value of tan 75°?
For the function y = 3cos(3θ) + 1, what is the maximum value?
For the function y = 3cos(3θ) + 1, what is the maximum value?
If csc θ = 2, what is sec(θ - π/2)?
If csc θ = 2, what is sec(θ - π/2)?
What is the phase shift of the graph of y = csc(2x) + 2?
What is the phase shift of the graph of y = csc(2x) + 2?
For the function y = tan(θ) - 1, what is the vertical asymptote?
For the function y = tan(θ) - 1, what is the vertical asymptote?
What is the general approach to finding the reference angle for an angle in standard position?
What is the general approach to finding the reference angle for an angle in standard position?
In a right triangle, if the lengths of the two legs are given as 'a' and 'b', what value directly corresponds to the cosecant of angle A?
In a right triangle, if the lengths of the two legs are given as 'a' and 'b', what value directly corresponds to the cosecant of angle A?
Which value represents an undefined function in trigonometric terms?
Which value represents an undefined function in trigonometric terms?
In the function $y = 1 + 3 an(2 heta)$, how does the period of the tangent function change?
In the function $y = 1 + 3 an(2 heta)$, how does the period of the tangent function change?
What is the relationship between the cosecant and sine functions?
What is the relationship between the cosecant and sine functions?
What is the behavior of the graph of $y = an( heta - 1)$ compared to $y = an( heta)$?
What is the behavior of the graph of $y = an( heta - 1)$ compared to $y = an( heta)$?
In evaluating $csc(780°)$, how would you simplify this angle to find an equivalent angle in the unit circle?
In evaluating $csc(780°)$, how would you simplify this angle to find an equivalent angle in the unit circle?
What can be inferred if the cosecant of an angle $ heta$ equals 2?
What can be inferred if the cosecant of an angle $ heta$ equals 2?
For the trigonometric function $y = 3 ext{cos}(2 heta) + 1$, what is the amplitude of the wave?
For the trigonometric function $y = 3 ext{cos}(2 heta) + 1$, what is the amplitude of the wave?
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Study Notes
Finding Reference Angles
- For angles greater than 360°: Subtract 360° repeatedly until the angle is between 0° and 360°.
- For negative angles: Add 360° repeatedly until the angle is between 0° and 360°.
- For angles in radians: Subtract 2𝜋 repeatedly until the angle is between 0 and 2𝜋.
Finding Trigonometric Ratios in a Right Triangle
- SOH CAH TOA:
- Sine: Opposite side / Hypotenuse
- Cosine: Adjacent side / Hypotenuse
- Tangent: Opposite side / Adjacent side
- Cosecant: 1 / Sine
- Secant: 1 / Cosine
- Cotangent: 1 / Tangent
- Pythagorean Theorem: a² + b² = c²
Evaluating Trigonometric Functions
- Unit Circle: Use the coordinates of points on the unit circle to find trigonometric function values.
- Special Angles: Know the trigonometric values for 30°, 45°, and 60°.
- Trigonometric Identities: Use identities to simplify expressions.
- Undefined Values: Trigonometric functions can be undefined at certain angles where the denominator is zero.
Graphing Trigonometric Functions
- Sine: Starts at the origin, oscillates between -1 and 1, and has a period of 2𝜋.
- Cosine: Starts at its maximum, oscillates between -1 and 1, and has a period of 2𝜋.
- Tangent: Has vertical asymptotes, oscillates between negative and positive infinity, and has a period of 𝜋.
- Cosecant: The reciprocal of sine, with vertical asymptotes at the zeros of sine.
- Secant: The reciprocal of cosine, with vertical asymptotes at the zeros of cosine.
- Cotangent: The reciprocal of tangent, with vertical asymptotes at the zeros of tangent.
- Amplitude: The height of the wave.
- Period: The length of one cycle of the function.
- Phase Shift: The horizontal shift of the function.
- Vertical Shift: The vertical shift of the function.
Finding Trigonometric Equations from Graphs
- Identify the amplitude, period, phase shift, and vertical shift.
- Determine whether the graph is a sine or cosine function.
- Write the equation using the general form:
- For sine: y = A sin(B(x - C)) + D
- For cosine: y = A cos(B(x - C)) + D
Evaluating Composite Trigonometric Functions
- Work from the inside out: Evaluate the innermost function first.
- Use the unit circle or special angles to find the trigonometric values.
- Simplify the expression using trigonometric identities.
Finding Reference Angles
- Calculate the reference angle for 215 degrees.
- Calculate the reference angle for -200 degrees.
- Calculate the reference angle for 7pi/6 radians.
Right Triangle Trigonometry
- Find the cosecant of angle A in a right triangle ABC where C is the right angle, given side a = 5 and side b = 2√6.
Trigonometric Function Values
- Find the exact value of the cosecant of 780 degrees or state if it is undefined.
- Find the exact value of the sine of 11pi/6 radians or state if it is undefined.
- Find the exact value of the secant of -pi/2 radians or state if it is undefined.
- Find the exact value of the tangent of 75 degrees.
- Find the exact value of the cosine of (11pi/6) radians.
- Find the exact value of the sine of (cosine (pi/3) + cosine (sin (pi/6))).
Graphing Trigonometric Functions
- Graph at least one period of the function y = 1 + 3cos(3theta + pi/3).
- Graph at least one period of the function y = 1/2csc(2x) + 2.
- Graph at least one period of the function y = tan(x - pi/4).
Graphing & Finding Equation
- Given a graph of a trigonometric function, find an equation of the function in either sine or cosine form.
Trigonometric Value Relationships
- Given that the cosecant of theta is 2, find the secant of (pi/2 - theta).
Evaluating Trigonometric Expressions
- Evaluate the expression csc(tan(pi/4)).
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