Trigonometric Values and Calculations

Trigonometric Values

• Sine and Cosine:
  • ( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} )
  • ( \cos \frac{\pi}{3} = \frac{1}{2} )

Sine, Cosine, and Tangent Calculations

• Given ( \sec x = \frac{13}{5} ), calculate:
  • Since ( \sec x = \frac{1}{\cos x} ), then ( \cos x = \frac{5}{13} ).
  • Use ( \sin^2 x + \cos^2 x = 1 ) to find ( \sin x = -\frac{12}{13} ) (as ( -\frac{\pi}{2} < x < \frac{\pi}{2} )).
  • Calculate ( \sin 2x = 2 \sin x \cos x = 2 \left(-\frac{12}{13}\right) \left(\frac{5}{13}\right) = -\frac{120}{169} ).
  • Calculate ( \cos 2x = 2 \cos^2 x - 1 = 2 \left(\frac{5}{13}\right)^2 - 1 = -\frac{119}{169} ).
  • Calculate ( \tan 2x = \frac{\sin 2x}{\cos 2x} = \frac{-\frac{120}{169}}{-\frac{119}{169}} = \frac{120}{119} ).

Trigonometric Proofs

• Proof i:

  • Prove ( \frac{1 - \cos 2\theta}{1 + \cos 2\theta} = \tan^2 \theta ) via identities.

• Proof ii:

  • Establish ( (\sin^2 t - \sin^2 x)\sin(x+t) + (\cos^2 t - \cos^2 x)\cos(x+t) = 0 ) using trigonometric identities.

• Proof iii & iv:

  • Show ( (cos^2 x - cos^2 y)^2 + (sin^2 x - sin^2 y)^2 = 4\sin^2(\frac{x+y}{2})\sin^2(\frac{x-y}{2}) ), using the Pythagorean identities.

• Proof v:

  • Demonstrate ( \tan x + \cot x = 2 \csc 2x ) with the addition formulas for tangent and cotangent.

• Proof vi:

  • Prove ( \frac{\cos x + \sin x}{\cos x - \sin x} - \frac{\cos x - \sin x}{\cos x + \sin x} = 2\tan 2x ) through algebraic manipulation.

• Proof vii:

  • Establish ( \sqrt{2 + \sqrt{2 + \sqrt{2 + \cos x}}} = 2 \cos(\frac{x}{8}) ) using nested radical properties.

• Proof viii:

  • Prove ( 16 \sin x \cos x \cos 2x \cos 4x \cos 8x = \sin 16x ) using product-to-sum formulas.

• Proof ix:

  • Show ( \frac{\sin 3x + \cos 3x}{\sin x + \cos x} = 2 \cos 2x ) using angle addition formulas and simplifications.

This quiz covers the fundamental concepts of trigonometric values, including sine, cosine, and tangent calculations. You will find specific examples and proofs related to these functions. Test your understanding of the relationships and identities in trigonometry.