Comprehensive Trigonometry Review Notes

Definitions of Trigonometric Functions

For a point P(x, y) on the terminal side of an angle q:

1. Define r: r = \text{distance from origin to P}

2. Sine Function:

   • \sin q = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}

   • \csc q = \frac{r}{y} = \frac{1}{\sin q}

3. Cosine Function:

   • \cos q = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}

   • \sec q = \frac{r}{x} = \frac{1}{\cos q}

4. Tangent Function:

   • \tan q = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x} = \frac{\sin q}{\cos q}

   • \cot q = \frac{x}{y} = \frac{1}{\tan q} = \frac{\cos q}{\sin q}

Measuring Angles

1. Circle Measures:

   • A circle contains 360^{\circ} or 2\pi radians.

2. Conversion Formula:

   • 180^{\circ} = \pi \text{ radians}

Example: Converting Degrees to Radians

Convert 36^{\circ} to radians:

1. Apply Conversion:

   • 36^{\circ} = 36^{\circ} \cdot \frac{\pi}{180^{\circ}} = \frac{2\pi}{10} = \frac{\pi}{5} \text{ radians}

Reference Angles

1. Definition:

   • The reference angle is formed by the terminal side of the angle and the x-axis.

2. Quadrant Calculations:

   • Quadrant II: R = 180^{\circ} - q

   • Quadrant III: R = q - 180^{\circ}

   • Quadrant IV: R = 360^{\circ} - q

Signs of Trigonometric Functions by Quadrant

1. Quadrant I:

   • All trigonometric functions are positive.

2. Quadrant II:

   • Sine is positive.

3. Quadrant III:

   • Tangent is positive.

4. Quadrant IV:

   • Cosine is positive.

5. Mnemonic:

   • "All Students Take Calculus" (

6. Reciprocal Signs:

   • Sine and cosecant have the same sign.

   • Cosine and secant have the same sign.

   • Tangent and cotangent have the same sign.

Special Triangles

1. Common Angles:

   • Using the ratios of common triangles with angles 30^{\circ}, 45^{\circ}, and 60^{\circ} allows us to determine the trigonometric functions of these special angles.

2. Values:

   • \sin 45^{\circ} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}

   • \sin 30^{\circ} = \frac{1}{2}

   • \cos 45^{\circ} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}

   • \cos 30^{\circ} = \frac{\sqrt{3}}{2}

   • \tan 45^{\circ} = \frac{1}{1} = 1

   • \tan 30^{\circ} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}

Trigonometry Review - Workshop 2

Unit Circle Definition

1. Sine and Cosine:

   • If (x, y) is a point on the unit circle, then \cos q = x and \sin q = y.

2. Values for Multiples of 90°:

   • \cos 0^{\circ} = 1

   • \cos 90^{\circ} = 0

   • \cos 180^{\circ} = -1

   • \cos 270^{\circ} = 0

   • \sin 0^{\circ} = 0

   • \sin 90^{\circ} = 1

   • \sin 180^{\circ} = 0

   • \sin 270^{\circ} = -1

3. Tangent Values:

   • \tan 0^{\circ} = 0

   • \tan 90^{\circ} = \text{undefined}

4. Cotangent Values:

   • \cot 0^{\circ} = \text{undefined}

   • \cot 90^{\circ} = 0

5. Secant Values:

   • \sec 0^{\circ} = 1

   • \sec 90^{\circ} = \text{undefined}

6. Cosecant Values:

   • \csc 0^{\circ} = \text{undefined}

   • \csc 90^{\circ} = 1

7. Calculation Ability:

   • At this point, you should be able to calculate the trig functions of all angles with reference angles of 30^{\circ}, 45^{\circ}, 60^{\circ}, as well as all multiples of 90^{\circ}.

Trigonometry Review - Workshop 3

Major Trigonometric Identities

Pythagorean Identities

1. \sin^2 q + \cos^2 q = 1

2. 1 + \cot^2 q = \csc^2 q

3. \tan^2 q + 1 = \sec^2 q

4. Derivation:

   • The last two can be derived from the first by dividing by either \sin^2 q or \cos^2 q

Addition Formulas

1. \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

2. \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Double Angle Identities

1. Let A = B in the addition formulas:

   • \sin(2A) = 2 \sin A \cos A

   • \cos(2A) = \cos^2 A - \sin^2 A

Law of Cosines

1. a^2 = b^2 + c^2 - 2bc \cos A

Law of Sines

1. \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}

Trigonometry Review - Workshop 3

Graphs of Sine and Cosine Functions

1. From the Unit Circle:

   • \cos q = x and \sin q = y.

2. Sine Function:

   • As we move around the circle, y = \sin(x) measures the y-value of the point, starting at 0 and increasing to 1 at x = \frac{\pi}{2}.

3. Cosine Function:

   • y = \cos(x) measures the x-value of the point, starting at 1 and decreasing to 0 at x = \frac{\pi}{2}.

Key Values for Sine and Cosine

Transformations of Sine and Cosine Graphs

Amplitude

1. Definition:

   • The amplitude of y = A \sin(x) is |A|. A vertical stretch.

Period

1. Definition:

   • The period of y = \sin(Bx) is \frac{2\pi}{B}. A horizontal shrink.

Vertical Shift

1. Example:

   • For y = \sin(x) + 1, the graph is shifted vertically by 1 unit.

Horizontal Shift (Phase Shift)

1. Example:

   • For y = \cos(x - \frac{\pi}{2}), set x - \frac{\pi}{2} = 0 to find the shift.

   • Here, x = \frac{\pi}{2}, so the shift is to the right by \frac{\pi}{2}.

Trigonometry Review - Workshop 4

Solving Trigonometric Equations

Example: Solve \sin^2 x = 1 for all 0 \le x < 2\pi

1. Rearrange:

   • \sin^2 x - 1 = 0

2. Factor:

   • (\sin x + 1)(\sin x - 1) = 0

3. Solve:

   • \sin x = 1 or \sin x = -1

4. Solutions:

   • x = \frac{\pi}{2}, \frac{3\pi}{2}

Example: Solve \cos 2x = 1 for 0 \le x < 2\pi

Method 1: Using the Double Angle Formula

1. \cos 2x = \cos^2 x - \sin^2 x = 1

2. (1 - \sin^2 x) - \sin^2 x = 1

3. 1 - 2\sin^2 x = 1

4. \sin^2 x = 0 \implies \sin x = 0

5. Solutions:

   • x = 0, \pi

Method 2: Solving for 2x First

1. \cos 2x = 1 \implies 2x = 0

2. Add One Period:

   • 2x = 0 + 2\pi

3. Solve for x:

   • x = 0, \pi

Example: Solve \sin^2 3x = 1 for 0 \le x < 2\pi

1. \sin 3x = \pm 1

2. \sin 3x = 1 at 3x = \frac{\pi}{2}

3. \sin 3x = -1 at 3x = \frac{3\pi}{2}

4. Solve for x:

   • x = \frac{\pi}{6}, \frac{\pi}{2}

5. Period of \sin 3x:

   • The period of \sin 3x is \frac{2\pi}{3}, so add this to each solution until the values exceed 2\pi

6. Complete Set of Solutions:

   • x \in { \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6} }

More Identities

1. Importance:

   • Being able to identify the major ones, especially the double angle formulas and the famous \sin^2x + \cos^2x = 1 is very important.

Example: Prove \cos 2x + 2\sin^2 x = 1

1. Start with the more difficult side:

   • \cos 2x + 2\sin^2 x = (\cos^2 x - \sin^2 x) + 2\sin^2 x

2. = \cos^2 x + (-\sin^2 x + 2\sin^2 x)

3. = \cos^2 x + \sin^2 x = 1

Example: Prove (\sec x)(\cot x)(\csc x) = \csc^2 x

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