Math127 Properties Of The Trigonometric Function

Domain and Range of Trigonometric Functions

• Trigonometric Functions: Six primary trigonometric functions include sine, cosine, tangent, cosecant, secant, and cotangent.

  • Sine and Cosine:

    • Domain: All real numbers.

    • Range: [-1, 1].

  • Tangent:

    • Domain: All real numbers except odd multiples of ( \frac{\pi}{2} ) (e.g., ( \frac{\pi}{2}, \frac{3\pi}{2}, \ldots )).

    • Range: All real numbers.

  • Cosecant and Secant:

    • Domain: All real numbers except integral multiples of ( \pi ) (for cosecant) and odd multiples of ( \frac{\pi}{2} ) (for secant).

    • Range: All real numbers except values within the interval (-1, 1) for cosecant and outside (-1, 1) for secant.

  • Cotangent:

    • Domain: All real numbers except integral multiples of ( \pi ).

    • Range: All real numbers.

Periodic Properties

• Periodic Nature: Trigonometric functions are periodic, meaning they repeat values at regular intervals.

  • Period of Functions:

    • Sine and Cosine: Period of ( 2\pi ).

    • Tangent and Cotangent: Period of ( \pi ).

    • Cosecant and Secant: Period of ( 2\pi ).

  • Example of Periodicity:

    • ( \theta + 2\pi n ) (where ( n ) is an integer): Trigonometric function values remain unchanged.

    • Visual representation: A circle illustrates the relationship between angles and values of functions.

Fundamental Trigonometric Identities

• Reciprocal Identities:

  • Cosecant: ( csc(\theta) = \frac{1}{sin(\theta)} )

  • Secant: ( sec(\theta) = \frac{1}{cos(\theta)} )

  • Cotangent: ( cot(\theta) = \frac{1}{tan(\theta)} )

• Quotient Identity:

  • Tangent: ( tan(\theta) = \frac{sin(\theta)}{cos(\theta)} )

  • Cotangent: ( cot(\theta) = \frac{cos(\theta)}{sin(\theta)} )

• Pythagorean Identities:

  • Fundamental Identity: ( sin^2(\theta) + cos^2(\theta) = 1 )

  • Other Identities:

    • ( 1 + tan^2(\theta) = sec^2(\theta) )

    • ( 1 + cot^2(\theta) = csc^2(\theta) )

Even and Odd Functions

• Even Functions:

  • Cosine: ( cos(-\theta) = cos(\theta) )

  • Cosecant: ( csc(-\theta) = csc(\theta) )

• Odd Functions:

  • Sine: ( sin(-\theta) = -sin(\theta) )

  • Tangent: ( tan(-\theta) = -tan(\theta) )

  • Cotangent: ( cot(-\theta) = -cot(\theta) )

  • Secant: ( sec(-\theta) = sec(\theta) )

Reference Angles

• Definition: The acute angle formed by the terminal side of the angle and the x-axis.

• Finding Reference Angles:

  • First Quadrant: Angle itself.

  • Second Quadrant: ( 180° - \text{given angle} )

  • Third Quadrant: ( \text{given angle} - 180° )

  • Fourth Quadrant: ( 360° - \text{given angle} )

Example Calculations

• Finding Sine of ( \frac{17\pi}{4} ):

  • Convert to mixed numbers: ( 17\pi/4 = 4\pi + \frac{\pi}{4} )

  • Apply periodic property: ( sin(17\pi/4) = sin(\pi/4) = \frac{\sqrt{2}}{2} )

• Finding Cosine of ( 5\pi ):

  • Convert to circles: ( 5\pi = 4\pi + \pi )

  • Apply periodic property: ( cos(5\pi) = cos(\pi) = -1 )

• Finding Tangent of ( \frac{5\pi}{4} ):

  • Convert to degrees and reference angle: ( \frac{5\pi}{4} = 45° + 180° )

  • Referencing quadrant values: ( tan(\frac{5\pi}{4}) = 1 \rightarrow -\frac{\sqrt{2}}{2} )

Quadrant Identification Based on Function Values

• Example: If ( sin(\theta) < 0 ) and ( cos(\theta) < 0 ):

  • Angle is located in the third quadrant.

Exact Values Calculation and Identities

• Tangent Value Calculation:

  • Given ( \sin = \frac{2}{5} \), ( \cos = \frac{\sqrt{21}}{5} \)

  • Result: ( tan(\theta) = \frac{2}{\sqrt{21}} ) ⇒ Rationalize: ( \frac{2\sqrt{21}}{21} )

• Finding All Trigonometric Function Values:

  • Cosecant: ( csc(\theta) = \frac{5}{2} )

  • Secant: ( sec(\theta) = \frac{5}{\sqrt{21}} )

  • Cotangent: Reciprocal of tangent.

Example Problems and Solutions

• Example A: Find exact values using identities.

  • ( tan(20°) - \frac{sin(20°)}{cos(20°)} = tan(20°) - tan(20°) = 0 )

• Example B: Quadrant observation with trigonometric function signs.

  • Identifying inverses and rationalizing as needed.

• Using Even/Odd Properties:

  • ( sin(-45°) = -sin(45°) = -\frac{\sqrt{2}}{2} )

  • ( cos(-\pi) = cos(\pi) = -1 )

  • ( tan(-\frac{37\pi}{4}) = -tan(\frac{37\pi}{4}) = -1 )

Domain and Range of Trigonometric Functions

• Trigonometric Functions: Sine, cosine, tangent, cosecant, secant, cotangent.

• Sine and Cosine: Domain: All real numbers; Range: [-1, 1].

• Tangent: Domain: All reals except odd multiples of (\frac{\pi}{2}); Range: All reals.

• Cosecant and Secant: Domain: All reals except multiples of (\pi) for cosecant and odd multiples of (\frac{\pi}{2}) for secant; Range: All reals except (-1, 1) for cosecant and outside (-1, 1) for secant.

• Cotangent: Domain: All reals except multiples of (\pi); Range: All reals.

Periodic Properties

• Period: Sine/Cosine: 2\pi; Tangent/Cotangent: \pi; Cosecant/Secant: 2\pi.

• Example: (\theta + 2\pi n) = constant values.

Fundamental Trigonometric Identities

• Reciprocal Identities: [csc(\theta) = \frac{1}{sin(\theta)}, sec(\theta) = \frac{1}{cos(\theta)}, cot(\theta) = \frac{1}{tan(\theta)}]

• Quotient Identity: [tan(\theta) = \frac{sin(\theta)}{cos(\theta)}, cot(\theta) = \frac{cos(\theta)}{sin(\theta)}]

• Pythagorean Identities: [sin^2(\theta) + cos^2(\theta) = 1].

Even and Odd Functions

• Even: cos(\theta) and csc(\theta).

• Odd: sin(\theta), tan(\theta), cot(\theta), sec(\theta).

Reference Angles

• Definition: Acute angle with x-axis.

• Finding Angles: 1st: angle; 2nd: (180° - angle); 3rd: (angle - 180°); 4th: (360° - angle).

Example Calculations

• sin(\frac{17\pi}{4}): Evaluate to find reference angle.

• cos(5\pi): Convert, use periodic property.

• tan(\frac{5\pi}{4}): Reference angle method.

Quadrant Identification

• Example: If (sin(\theta) < 0 and cos(\theta) < 0), angle in 3rd quadrant.

Exact Values Calculation

• tan(\theta) from given sin and cos values, rationalize.

• Finding All Trig Values: Using relations between functions and identities.