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Trigonometry for AC Circuits
23 questions By Tony R. Kuphaldt
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Question 1 of 23
Identify which trigonometric functions (sine, cosine, or tangent) are represented by each of the following ratios, with reference to the angle labeled with the Greek letter “Theta” (Θ):
X R= X Z= R Z= Reveal answer
X R= tanΘ = Opposite AdjacentX Z= sinΘ = Opposite HypotenuseR Z= cosΘ = Adjacent HypotenuseNotes:Ask your students to explain what the words “hypotenuse”, “opposite”, and “adjacent” refer to in a right triangle.
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Question 2 of 23
Identify which trigonometric functions (sine, cosine, or tangent) are represented by each of the following ratios, with reference to the angle labeled with the Greek letter “Phi” (φ):
R X= X Z= R Z= Reveal answer
R X= tanφ = Opposite AdjacentX Z= cosφ = Adjacent HypotenuseR Z= sinφ = Opposite HypotenuseNotes:Ask your students to explain what the words “hypotenuse”, “opposite”, and “adjacent” refer to in a right triangle.
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Question 3 of 23
The impedance triangle is often used to graphically relate Z, R, and X in a series circuit:
Unfortunately, many students do not grasp the significance of this triangle, but rather memorize it as a “trick” used to calculate one of the three variables given the other two. Explain why a right triangle is an appropriate form to relate these variables, and what each side of the triangle actually represents.
Reveal answerEach side of the impedance triangle is actually a phasor (a vector representing impedance with magnitude and direction):
Since the phasor for resistive impedance (ZR) has an angle of zero degrees and the phasor for reactive impedance (ZC or ZL) either has an angle of 90 or -90 degrees, the phasor sum representing total series impedance will form the hypotenuse of a right triangle when the first to phasors are added (tip-to-tail).
Follow-up question: as a review, explain why resistive impedance phasors always have an angle of zero degrees, and why reactive impedance phasors always have angles of either 90 degrees or -90 degrees.
Notes:The question is sufficiently open-ended that many students may not realize exactly what is being asked until they read the answer. This is okay, as it is difficult to phrase the question in a more specific manner without giving away the answer!
