IB Mathematics Applications & Interpretation Topic AHL 3.8—unit Circle, Pythag Identity, Solving Trig Equations Graphically Revision Notes (HL) | RevisionDojo

Unit Circle Definitions and Exact Trigonometric Ratios

The Unit Circle

The unit circle is a circle with radius 1 centered at the origin (0,0).

On the unit circle:

The x-coordinate is $\cos \theta$
The y-coordinate is $\sin \theta$
The slope of the radius line is $\tan \theta = \frac{\sin \theta}{\cos \theta}$

Tip

Remember that any point on the unit circle can be written as $(\cos \theta, \sin \theta)$ because the radius is always 1!

Unit Circle and Ratios

Depending on which quadrant $\theta$ is in, the values of $\sin \theta, \cos \theta, \tan \theta $ may change sign.

Since $\sin\theta$ is correlated with the $y$-value of the point on the unit circle, $\sin\theta$ is positive in the first and second quadrants ($y>0$), and negative in the third and fourth ($y<0$).
Since $\cos\theta$ is correlated with the $x$-value of the point on the unit circle, $\cos\theta$ is positive in the first and fourth quadrants ($x>0$), and negative in the second and third ($x<0$).
Since $\tan\theta$ is correlated with the slope of the line to the point on the unit circle, $\tan\theta$ is positive in the first and third quadrants ($y/x > 0$) and negative in the second and fourth quadrants ($y/x < 0$).

Tip

Some students use the mnemonic "All Students Take Calculus" for which trigonometric functions are positive in which quadrants.

All functions are positive in the first quadrant.
Sine is positive in the second quadrant.
Tangent is positive in the third quadrant.
Cosine is positive in the fourth quadrant.

The Ambiguous Case of the Sine Rule

The sine rule states: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

When using the sine rule to solve for an angle, we sometimes get two possible solutions. This is called the ambiguous case.

Example

If we know:

Side a = 8
Angle A = 40°
Side b = 7

When solving for angle B, we get: $\sin B = \frac{7 \sin 40°}{8}$

This could give us two angles:

B = 34.9°
B = 145.1°

Both are valid solutions unless we have additional information!

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Note

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Tip

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Unit Circle Definitions and Exact Trigonometric Ratios

The Unit Circle

The unit circle is a circle with radius 1 centered at the origin (0,0).

On the unit circle:

The x-coordinate is $\cos \theta$
The y-coordinate is $\sin \theta$
The slope of the radius line is $\tan \theta = \frac{\sin \theta}{\cos \theta}$

Tip

Remember that any point on the unit circle can be written as $(\cos \theta, \sin \theta)$ because the radius is always 1!

Unit Circle and Ratios

Depending on which quadrant $\theta$ is in, the values of $\sin \theta, \cos \theta, \tan \theta $ may change sign.

Since $\sin\theta$ is correlated with the $y$-value of the point on the unit circle, $\sin\theta$ is positive in the first and second quadrants ($y>0$), and negative in the third and fourth ($y<0$).
Since $\cos\theta$ is correlated with the $x$-value of the point on the unit circle, $\cos\theta$ is positive in the first and fourth quadrants ($x>0$), and negative in the second and third ($x<0$).
Since $\tan\theta$ is correlated with the slope of the line to the point on the unit circle, $\tan\theta$ is positive in the first and third quadrants ($y/x > 0$) and negative in the second and fourth quadrants ($y/x < 0$).

Tip

Some students use the mnemonic "All Students Take Calculus" for which trigonometric functions are positive in which quadrants.

All functions are positive in the first quadrant.
Sine is positive in the second quadrant.
Tangent is positive in the third quadrant.
Cosine is positive in the fourth quadrant.

The Ambiguous Case of the Sine Rule

The sine rule states: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

When using the sine rule to solve for an angle, we sometimes get two possible solutions. This is called the ambiguous case.

Example

If we know:

Side a = 8
Angle A = 40°
Side b = 7

When solving for angle B, we get: $\sin B = \frac{7 \sin 40°}{8}$

This could give us two angles:

B = 34.9°
B = 145.1°

Both are valid solutions unless we have additional information!

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Note

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Excepteur sint occaecat cupidatat non proident

Tip

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Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics

IB Mathematics Applications & Interpretation AHL 3.8—unit Circle, Pythag Identity, Solving Trig Equations Graphically Notes

Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics

Unit Circle Definitions and Exact Trigonometric Ratios

The Unit Circle

Unit Circle and Ratios

The Ambiguous Case of the Sine Rule

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Unit Circle Definitions and Exact Trigonometric Ratios

The Unit Circle

Unit Circle and Ratios

The Ambiguous Case of the Sine Rule

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account