Trigonometric Identities Quiz

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}} )

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)

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)} )

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 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) )

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 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}} )

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 )

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} ).

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.