Trigonometric graphs, identities and general triangles
| English | Português |
|---|---|
| amplitude/ˈæmplɪtjuːd/ | amplitude |
| included angle/ɪnˈkluːdɪd ˈæŋɡl/ | ângulo incluído |
A decision before an answer
- SOHCAHTOA uses a right triangle. A non-right triangle needs an appropriate general-triangle relationship.
- Your goal: Interpret amplitude, period and vertical shift in a sinusoidal rule.
Read the relationship
- For a right triangle, sin is opposite/hypotenuse, cos adjacent/hypotenuse and tan opposite/adjacent relative to a chosen angle. Similar triangles explain why these ratios depend on angle rather than size. On the unit circle, coordinates are (cos θ,sin θ), extending the ratios to other quadrants with appropriate signs.
- Use unit-circle signs and basic identities.
y=2 sin(3x)+4 has maximum:
Midline 4 plus amplitude 2 is 6.
Use the defining rule
- sin²θ+cos²θ=1 and tan θ=sin θ/cos θ when cos θ≠0. For y=A sin(Bx)+D with radian x, amplitude is |A|, period 2π/|B| and midline D. A negative A reflects the wave; D shifts it vertically. Check whether the input is in radians or degrees before using a graph interval.
- Choose a sine, cosine or area relation for a general triangle.
For radian x, sin(2x) has period:
2π/2=π.
Check the conditions
- For a general triangle, the cosine rule c²=a²+b²-2ab cos C uses the included angle C between a and b. The sine rule compares each side with the sine of its opposite angle. Area is ab sin C/2 when two sides and their included angle are known. Choose the relationship matching the supplied information.
- Choose a sine, cosine or area relation for a general triangle.
y=3 sin(2x)+1 has amplitude 3, period π and values between -2 and 4. A triangle with sides 5 and 7 around a 60° angle has third side √(25+49-70·1/2)=√39 and area 35√3/4. At 150°, sine is 1/2 and cosine -√3/2.
sin²θ+cos²θ=____.
This follows from the unit-circle equation.
Apply the task format
- Do not infer a right angle from the sketch. A sine-based equation may allow more than one angle in a permitted interval; retain the quadrant information and any triangle sum condition. Keep amplitude distinct from total peak-to-trough distance, which is twice the amplitude.
- Choose a sine, cosine or area relation for a general triangle.
State angle units and identify the included angle. The highest value is midline plus amplitude, not the amplitude alone.
Which answer fits this case?
Interpret amplitude, period and vertical shift in a sinusoidal rule
The Pythagorean theorem directly applies to every triangle.
It requires a right angle.
Keep the distinctions
- amplitude 振幅 — The distance from a sinusoid’s midline to a peak.
- included angle 夹角 — The angle between the two specified sides.
- Interpret amplitude, period and vertical shift in a sinusoidal rule.
- Use unit-circle signs and basic identities.
- Choose a sine, cosine or area relation for a general triangle.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.