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Proofs of trigonometric identities

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Proofs of trigonometric identities are used to show relations between trigonometric functions. This article will list trigonometric identities and prove them.


Elementary trigonometric identities

Ratio identities

Trigonometric Triangle

Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle. The sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The cosine of angle θ is defined as being the length of the adjacent side divided by the length of the hypotenuse. The tangent of θ is defined as being the length of the opposite side divided by the length of the adjacent side. The cosecant of θ is defined as being the length of the hypotenuse divided by the length of the opposite side. The secant of θ is defined as being the length of the hypotenuse side divided by the length of the adjacent side. The cotangent of θ is defined as being the length of the adjacent side divided by the length of the opposite side.

Tangent

 \tan \theta = \frac {\sin \theta}{\cos \theta}

Proof:

  1. \frac{\sin \theta}{\cos \theta} = \frac{\frac{opposite}{hypotenuse}}{\frac{adjacent}{hypotenuse}} (definition of the sine and cosine)
  2. \frac{\sin \theta}{\cos \theta} = \frac{\frac{opposite}{hypotenuse} \frac{hypotenuse}1}{\frac{adjacent}{hypotenuse} \frac{hypotenuse}1} (multiply numerator and denominator by hypotenuse)
  3. \frac{\sin \theta}{\cos \theta} = \frac{opposite}{adjacent} = \tan \theta (definition of tangent)

Q.E.D.

 \tan \theta = \frac {\sec \theta}{\csc \theta}

Proof:

  1. \frac {\sec \theta}{\csc \theta} = \frac{\frac{hypotenuse}{adjacent}}{\frac{hypotenuse}{opposite}} (definition of secant and cosecant)
  2. \frac{\sec \theta}{\csc \theta} = \frac{\frac{hypotenuse}{adjacent} \frac{1}{hypotenuse}}{\frac{hypotenuse}{opposite}\frac{1}{hypotenuse}} (multiply numerator and denominator by \frac{1}{hypotenuse})
  3. \frac {\sec \theta}{\csc \theta} = \frac{\frac{1}{adjacent}}{\frac{1}{opposite}} (Simplify)
  4. \frac {\sec \theta}{\csc \theta} = \frac{opposite}{adjacent} = \tan \theta (definition of the tangent)

Q.E.D.

Cotangent

 \cot \theta = \frac {\cos \theta}{\sin \theta}

Proof:

  1. \frac{\cos \theta}{\sin \theta} = \frac{\frac{adjecent}{hypotenuse}}{\frac{opposite}{hypotenuse}} (definition of the sine and cosine)
  2. \frac{\cos \theta}{\sin \theta} = \frac{\frac{adjacent}{hypotenuse} \frac{hypotenuse}1}{\frac{opposite}{hypotenuse} \frac{hypotenuse}1} (multiply numerator and denominator by hypotenuse)
  3. \frac{\cos \theta}{\sin \theta} = \frac{adjacent}{opposite} = \cot \theta (definition of cotangent)

Q.E.D.

 \cot \theta = \frac {\csc \theta}{\sec \theta}

Proof:

  1. \frac {\csc \theta}{\sec \theta} = \frac{\frac{hypotenuse}{opposite}}{\frac{hypotenuse}{adjacent}} (definition of secant and cosecant)
  2. \frac{\csc \theta}{\sec \theta} = \frac{\frac{hypotenuse}{opposite} \frac{1}{hypotenuse}}{\frac{hypotenuse}{adjacent}\frac{1}{hypotenuse}} (multiply numerator and denominator by \frac{1}{hypotenuse})
  3. \frac {\sec \theta}{\csc \theta} = \frac{\frac{1}{opposite}}{\frac{1}{adjacent}} (Simplify)
  4. \frac {\sec \theta}{\csc \theta} = \frac{adjacent}{opposite} = \cot \theta (definition of the cotangent)

Q.E.D.

Inequalities

Sine

\frac{\sin \theta}{\theta} < 1

Proof: Template:Sectstub

Tangent

\frac{\tan \theta}{\theta} > 1

Proof: Template:Sectstub

Pythagorean identities

Basic Pythagorean identity

\sin^2(x) + \cos^2(x) = 1 \!

Proof:

  1. a^2 + b^2 = h^2 \!\ (Pythagorean Theorem)
  2. a = h \sin \theta \!\ and b = h \cos \theta \!\ (definition of sine and cosine)
  3. a^2 = h^2 \sin^2 \theta \!\ and b^2 = h^2 \cos^2 \theta \!\ (square both sides of each equation)
  4. h^2 \sin^2(x) + h^2 \cos^2(x) = h^2 \!\ (substitute expressions for a² and b² into first equation)
  5. \sin^2(x) + \cos^2(x) = 1 \! (Divide both sides by h²)

Q.E.D.

Tangent and Secant

\tan^2 \theta + 1 = \sec^2 \theta \!\

Proof:

  1. \sin^2 \theta + \cos^2 \theta = 1 \!\ (Basic Pythagorean identity)
  2. \frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac 1{\cos^2 \theta} (divide both sides by \cos^2 \theta \ )
  3. \tan^2 \theta + 1 = \sec^2 \theta \!\ (tangent ratio and definition of secant)

Q.E.D.

Cotangent and Cosecant

\cot^2 \theta + 1 = \csc^2 \theta \!\

Proof:

  1. \cos^2 \theta + \sin^2 \theta = 1 \!\ (Basic Pythagorean identity)
  2. \frac{\cos^2 \theta}{\sin^2 \theta} + \frac{\sin^2 \theta}{\sin^2 \theta} = \frac 1{\sin^2 \theta} (divide both sides by \sin^2 \theta \ )
  3. \cot^2 \theta + 1 = \csc^2 \theta \!\ (cotangent ratio and definition of cosecant)

Q.E.D.

Identities involving Calculus

Sine and angle ratio identity

\lim_{\theta \to 0}{\frac{\sin \theta}{\theta}} = 1

Proof:

  1.  \frac{\tan \theta}{\theta} > 1 (tangent inequality identity)
  2.  \frac{\tan \theta}{\theta} = \frac{\sin \theta}{\theta \cos \theta} > 1 (tanget ratio identity)
  3.  \frac{\sin \theta}{\theta} > \cos \theta (multiply both sides by \cos \theta \!\ )
  4.  1 > \frac{\sin \theta}{\theta} > \cos \theta (sine inequality identity)
  5.  \lim_{\theta \to 0} 1 = 1 and  \lim_{\theta \to 0} \cos \theta = 1 (\cos 0 = 0 \ )
  6. \lim_{\theta \to 0}{\frac{\sin \theta}{\theta}} = 1 (Squeeze Theorem)

Q.E.D.

Cosine and angle ratio identity

\lim_{\theta \to 0}\frac{\cos \theta - 1}{\theta} = 0

Proof:

  1.  \lim_{\theta \to 0}\frac{\cos \theta - 1}{\theta} = \lim_{\theta \to 0}\frac{\cos^2 \theta - 1}{\theta (\cos \theta + 1)} (multiply numerator and denominator by  \cos \theta + 1 \ )
  2.  \cos^2(x) - 1 = -\sin^2(x)\ (Basic Pythagorean identity)
  3. \lim_{\theta \to 0}\frac{\cos \theta - 1}{\theta} = \lim_{\theta \to 0}\frac{-\sin^2 \theta}{\theta(\cos \theta + 1)} (substitution)
  4. \lim_{\theta \to 0}\frac{\cos \theta - 1}{\theta} = -\lim_{\theta \to 0} \frac{\sin \theta}{\theta} \lim_{\theta \to 0} \frac{\sin \theta}{\cos \theta + 1} (multiplication of limits property)
  5. -\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = -1 (sine and angle ratio identity)
  6. \lim_{\theta \to 0} \frac{\sin \theta}{\cos \theta + 1} = \frac{0}{2} = 0 (since \sin 0 = 0 \ and \cos 0 = 1 \ )
  7. \lim_{\theta \to 0}\frac{\cos \theta - 1}{\theta}= (-1)(0) = 0 \ (from steps 5 and 6)

Q.E.D.

Cosine and square of angle ratio identity

 \lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta^2}  = \frac{1}{2}

Proof:

  1. (1 - \cos \theta)(1 + \cos \theta) = 1 - \cos^2 \theta \ (polynomial multiplication)
  2.  \lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta^2} = \lim_{\theta \to 0} \ \frac{1 - \cos^2 \theta}{\theta^2 \ (1 + \cos \theta)} (multiply both sides by (1 + \cos \theta)\,)
  3. 1 - \cos^2 \theta = \sin^2 \theta \ (basic Pythagorean identity)
  4.  \lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta^2}=\lim_{\theta \to 0} \ \frac{\sin^2 \theta}{\theta^2 \ (1 + \cos \theta)} (substitution)
  5.  \lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta^2}=\bigg(\lim_{\theta \to 0}\frac{\sin \theta}{\theta}\bigg)^2 \ \lim_{\theta \to 0}\frac{1}{1 + \cos \theta} (limit properties)
  6.  \bigg( \lim_{\theta \to 0}\frac{\sin \theta}{\theta} \bigg)^2 = 1 (sine and angle ratio identity)
  7.  \lim_{\theta \to 0}\frac{1}{1 + \cos \theta} = \frac 1 2 (since  \cos 0 = 1 \  )
  8.  \lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta^2}  = \frac{1}{2} (steps 6 and 7)

Q.E.D.

Identity 4:  \lim_{\phi \to 0}\frac{\sin k\phi}{p\phi} = \frac{k}{p}

Proof

  1. = \lim_{\phi \to 0}\frac{\sin k\phi}{p\phi} \frac{k}{k} (multiplicative identity)
  2. = \lim_{\phi \to 0}\frac{k}{p}\frac{\sin k\phi}{k\phi} (reorder variables in denominator)
  3. = \frac{k}{p} \lim_{\phi \to 0}\frac{\sin k\phi}{k\phi} (limit laws)
  4. \lim_{\phi \to 0}\frac{\sin k\phi}{k\phi} = \lim_{k\phi \to 0}\frac{\sin k\phi}{k\phi} Substituting θ for kφ, we get Identity 1. Thus,
  5. = \frac{k}{p} (1) = \frac{k}{p}

Q.E.D.

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