Unit 7 · TrigonometryG6 · Trigonometry · Aerodynamics & Navigation

Trig Graphs & Oblique Triangles

Read amplitude and period, then use the laws of sines and cosines when triangles are not right.

28 min
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Start with the warm-up.

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Warm-up

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Key terms

amplitude: Half the vertical distance from a sinusoid minimum to maximum.
period: The horizontal length of one full cycle.
included angle: The angle between two named sides.

Start by naming the quantity each problem asks for. That habit blocks the most common ACT trap.

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Take a calm first shot. Label what you know, write one relationship, and make the best choice you can.

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The big idea

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Build the picture before the arithmetic

First ask whether a right angle is guaranteed. For $y=A\sin(Bx)$ , the amplitude is $|A|$ and the period in degrees is $\frac{360^\circ}{|B|}$ . For oblique triangles, write the appropriate law before substituting.

Key idea

The decision that matters

Use the triangle information to choose the law: a known angle-opposite-side pair points to sines; two sides with their included angle points to cosines.

Law of cosines

c^2=a^2+b^2-2ab\cos C

Write this relationship before inserting numbers. It keeps labels and units attached to the work.

A compact example

If $a=7$ , $b=10$ , and $C=60^\circ$ , then $c^2=7^2+10^2-2(7)(10)\cos60^\circ=79$ , so $c=\sqrt{79}$ .

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Read amplitude and period from a trig graph

Keep degree mode on. Read the maximum and minimum y-values to get amplitude 4, then locate two consecutive maxima to get the 120° period. Change 3 to 2 and verify that the period becomes 180°.

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Read amplitude and period from a trig graph

Degree mode

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Cue	Best move
Trig graph	Use the outside coefficient for amplitude and the inside factor for period
Oblique triangle	SAS uses cosine; known opposite pairs use sine

Trap

Using SOH-CAH-TOA on a nonright triangle

This error produces a believable answer. Pause before computing and say what each number means.

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Worked examples

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Skip check · 2 questions

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Worked example

Example 1: Read a sinusoid

Problem

For

y=-4\cos(3x)

, with

x

in degrees, find the amplitude and period.

Amplitude is the absolute value of the outside coefficient:

|-4|=4

Period is $360^\circ/|3|=120^\circ$ .

What are the amplitude and period of this cosine graph?

The negative reflects the graph; it does not make amplitude negative.

Why do the coefficient 4 and factor 3 give those graph features?

Worked example

Example 2: Use the law of cosines

Problem

Two sides of a triangle are 8 and 11, and their included angle is

60^\circ

. Find the square of the opposite side

c

This is side-angle-side, so use

c^2=a^2+b^2-2ab\cos C

$c^2=8^2+11^2-2(8)(11)\cos60^\circ$ .

Which substitution correctly uses the included angle?

Since $\cos60^\circ=1/2$ , $c^2=64+121-88=97$ .

Why is the law of cosines appropriate here?

Faded example · you finish it

Example 3: You finish it

Problem

In a triangle, side 10 is opposite

40^\circ

and side

b

is opposite

65^\circ

. Complete the setup.

Match each side with its opposite angle before writing a sine-law proportion.

Which completion is correct?

Which reason validates the result?

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.