Trigonometry - Additional Topic - Quick Guides

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Study GuideTrigonometry–Additional Topic1. Simple Harmonic MotionMany motions in real life repeat in a regular pattern. Think about:•A swinging pendulum•A vibrating guitar string•Alternating current in electricityThese types of motion are calledsimple harmonic motion.1.1Description ofSimple Harmonic MotionSimple harmonic motion is described by equations like:Let’s break this down:•d= displacement (how far something is from its starting position)•A= amplitude (maximum displacement)•B= a constant related to frequency•t= timeSo in simple terms:The position of an object changes over time following a sine or cosine pattern.Because sine and cosine repeat, the motion repeats too.Example 1:Voltage in an Electric CurrentThe voltage in a current is given by:

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Study Guidewhere:•(E) is in volts•(t) is in secondsWe are asked to find (E) when:Use (π = 3.1416).Step 1: Substitute the Value of tStep 2: Reduce the AngleSince sine is periodic it repeats every (2π),we reduce 993.6 by subtracting multiples of (2π).After removing 158 full rotations of (2π),we are left with:So now we evaluate:Step 3: CalculateFinal Answer

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Study GuideExample 2:Motion of a PendulumThe horizontal displacement of a pendulum is:We are given:We need to find (K).Step 1: Substitute the ValuesUsing (π = 3.1416):Step 2: Evaluate the SineSo the equation becomes:Final AnswerThis means the amplitude (maximum displacement) of the pendulum is about 12 cm.

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Study GuideSummarySimple harmonic motion follows sine or cosine patterns.Amplitude (A or K) tells you the maximum displacement.The motion repeats because sine and cosine are periodic.You can reduce large angles using the fact that sine repeats every (2π).2. The Expression M sin Bt + N cos BtFigure 1 Reference graph fory=MsinBt+NcosBt.Sometimes a trigonometric function is written like this:At first, this looks complicated. But here’s the good news:Any expression of this form can be rewritten as a single sine function:So these two forms areequivalent. They describe the same motion—just written differently.This result comes from thesum identity for sine.How Are the Constants Related?If:

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Study Guidethen it can be rewritten as:where:This value is very important.Key RelationshipsLet:Then:From this, you can determine the angle (C).Important PropertiesOnce written in the form:we can easily identify:•Amplitude:

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Study Guide•Period:•Frequency:•Phase Shift:This makes the function much easier to analyze.Example 1:Convert:Figure 2 Drawing for Example 1.into the form:Then find the amplitude, period, frequency, and phase shift.

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