Trigonometric Identities Quiz

Trigonometric Identities

• Basic Identities

  • Pythagorean Identity: ( \sin^2(\theta) + \cos^2(\theta) = 1 )
  • Reciprocal Identities:
    • ( \csc(\theta) = \frac{1}{\sin(\theta)} )
    • ( \sec(\theta) = \frac{1}{\cos(\theta)} )
    • ( \cot(\theta) = \frac{1}{\tan(\theta)} )
  • Quotient Identities:
    • ( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} )
    • ( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} )

• Co-Function Identities

  • ( \sin(\frac{\pi}{2} - \theta) = \cos(\theta) )
  • ( \cos(\frac{\pi}{2} - \theta) = \sin(\theta) )
  • ( \tan(\frac{\pi}{2} - \theta) = \cot(\theta) )

• Even-Odd Identities

  • Even: ( \cos(-\theta) = \cos(\theta) )
  • Odd:
    • ( \sin(-\theta) = -\sin(\theta) )
    • ( \tan(-\theta) = -\tan(\theta) )

• Sum and Difference Formulas

  • ( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) )
  • ( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) )
  • ( \tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)} )

• Double Angle and Half Angle Formulas

  • Double Angle:
    • ( \sin(2\theta) = 2\sin(\theta)\cos(\theta) )
    • ( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) )
  • Half Angle:
    • ( \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}} )
    • ( \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} )

Angles and Their Measures

• Types of Angles

  • Acute Angle: ( 0^\circ < \theta < 90^\circ )
  • Right Angle: ( \theta = 90^\circ )
  • Obtuse Angle: ( 90^\circ < \theta < 180^\circ )
  • Straight Angle: ( \theta = 180^\circ )
  • Reflex Angle: ( 180^\circ < \theta < 360^\circ )

• Angle Measurement Units

  • Degrees: Full circle = ( 360^\circ )
  • Radians: Full circle = ( 2\pi ) radians (1 radian ( \approx 57.3^\circ ))
  • Conversion: ( \theta \text{ (degrees)} = \theta \text{ (radians)} \times \frac{180}{\pi} )

• Standard Position of Angles

  • Angles measured from the positive x-axis.
  • Counterclockwise rotation is positive; clockwise rotation is negative.

• Reference Angles

  • The acute angle formed by the terminal side of an angle and the x-axis.
  • Useful for determining the values of trigonometric functions in different quadrants.

• Quadrants

  • I: ( 0^\circ < \theta < 90^\circ ) (All functions positive)
  • II: ( 90^\circ < \theta < 180^\circ ) (Sine positive)
  • III: ( 180^\circ < \theta < 270^\circ ) (Tangent positive)
  • IV: ( 270^\circ < \theta < 360^\circ ) (Cosine positive)

Trigonometric Identities

• Basic Identities include the Pythagorean Identity ( \sin^2(\theta) + \cos^2(\theta) = 1 ), foundational for all trigonometric functions.
• Reciprocal Identities establish relationships between sine, cosine, and their reciprocal functions:
  • ( \csc(\theta) = \frac{1}{\sin(\theta)} )
  • ( \sec(\theta) = \frac{1}{\cos(\theta)} )
  • ( \cot(\theta) = \frac{1}{\tan(\theta)} )
• Quotient Identities define tangent and cotangent in terms of sine and cosine:
  • ( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} )
  • ( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} )

Co-Function Identities

• Pivotal relations between sine, cosine, and tangent for angles complementary to ( \theta ):
  • ( \sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) )
  • ( \cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta) )
  • ( \tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta) )

Even-Odd Identities

• Even Function: Cosine remains unchanged for negative inputs: ( \cos(-\theta) = \cos(\theta) ).
• Odd Functions: Sine and tangent change sign for negative inputs:
  • ( \sin(-\theta) = -\sin(\theta) )
  • ( \tan(-\theta) = -\tan(\theta) )

Sum and Difference Formulas

• Useful for calculating the sine, cosine, and tangent of angle sums or differences:
  • ( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) )
  • ( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) )
  • ( \tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)} )

Double Angle and Half Angle Formulas

• Double Angle Formulas express trigonometric functions of doubled angles:
  • ( \sin(2\theta) = 2\sin(\theta)\cos(\theta) )
  • ( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) )
• Half Angle Formulas provide expressions for half angles:
  • ( \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}} )
  • ( \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} )

Angles and Their Measures

• Types of Angles are classified based on their measures:
  • Acute: ( 0^\circ < \theta < 90^\circ )
  • Right: ( \theta = 90^\circ )
  • Obtuse: ( 90^\circ < \theta < 180^\circ )
  • Straight: ( \theta = 180^\circ )
  • Reflex: ( 180^\circ < \theta < 360^\circ )

Angle Measurement Units

• Angles can be measured in degrees and radians:
  • Full circle: ( 360^\circ ) in degrees, ( 2\pi ) radians in radians.
  • Conversion factor: ( \theta \text{ (degrees)} = \theta \text{ (radians)} \times \frac{180}{\pi} ).

Standard Position of Angles

• Angles are measured from the positive x-axis.
• Counterclockwise rotation is positive, while clockwise rotation is negative.
• Reference Angles: Acute angles with the x-axis, crucial for evaluating trigonometric functions across quadrants.
• Quadrants:
  • I: ( 0^\circ < \theta < 90^\circ ); all trig functions are positive.
  • II: ( 90^\circ < \theta < 180^\circ ); sine is positive.
  • III: ( 180^\circ < \theta < 270^\circ ); tangent is positive.
  • IV: ( 270^\circ < \theta < 360^\circ ); cosine is positive.

Test your understanding of basic trigonometric identities, including Pythagorean, reciprocal, co-function, and even-odd identities. Delve into sum and difference formulas to ensure mastery of important concepts in trigonometry. Perfect for students preparing for exams or those seeking to strengthen their math skills.