Solving for cos 2A using tan A

This problem asks us to find the value of \(\cos 2A\) given the value of \(\tan A\). We can use a standard trigonometric identity that relates \(\cos 2A\) directly to \(\tan A\).

Trigonometric Identity for cos 2A

There are several formulas for \(\cos 2A\), but the one most useful here is the identity expressed in terms of \(\tan A\). The identity is:

\[ \cos 2A = \dfrac{1 - \tan^2 A}{1 + \tan^2 A} \]

This identity allows us to calculate \(\cos 2A\) directly if we know \(\tan A\).

Step-by-Step Calculation of cos 2A

We are given that \(\tan A = \dfrac{1}{7}\).

First, we need to calculate \(\tan^2 A\):

\[ \tan^2 A = \left( \dfrac{1}{7} \right)^2 = \dfrac{1^2}{7^2} = \dfrac{1}{49} \]

Now, substitute this value into the identity for \(\cos 2A\):

\[ \cos 2A = \dfrac{1 - \tan^2 A}{1 + \tan^2 A} = \dfrac{1 - \dfrac{1}{49}}{1 + \dfrac{1}{49}} \]

To simplify this complex fraction, we can find a common denominator for the numerator and the denominator. The common denominator is 49.

Numerator: \(1 - \dfrac{1}{49} = \dfrac{49}{49} - \dfrac{1}{49} = \dfrac{49 - 1}{49} = \dfrac{48}{49}\)

Denominator: \(1 + \dfrac{1}{49} = \dfrac{49}{49} + \dfrac{1}{49} = \dfrac{49 + 1}{49} = \dfrac{50}{49}\)

Now substitute these back into the expression for \(\cos 2A\):

\[ \cos 2A = \dfrac{\dfrac{48}{49}}{\dfrac{50}{49}} \]

To divide fractions, we multiply the numerator by the reciprocal of the denominator:

\[ \cos 2A = \dfrac{48}{49} \times \dfrac{49}{50} \]

We can cancel out the 49 from the numerator and the denominator:

\[ \cos 2A = \dfrac{48}{50} \]

Finally, simplify the fraction \(\dfrac{48}{50}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

\[ \cos 2A = \dfrac{48 \div 2}{50 \div 2} = \dfrac{24}{25} \]

So, the value of \(\cos 2A\) is \(\dfrac{24}{25}\).

Trigonometry Revision Table

Additional Trigonometry Information

• Double Angle Formulas: These formulas express trigonometric functions of \(2A\) in terms of trigonometric functions of \(A\). Key identities for \(\cos 2A\) include:
  • \(\cos 2A = \cos^2 A - \sin^2 A\)
  • \(\cos 2A = 2 \cos^2 A - 1\)
  • \(\cos 2A = 1 - 2 \sin^2 A\)
  • \(\cos 2A = \dfrac{1 - \tan^2 A}{1 + \tan^2 A}\)
• Pythagorean Identities: Fundamental identities derived from the Pythagorean theorem:
  • \(\sin^2 A + \cos^2 A = 1\)
  • \(1 + \tan^2 A = \sec^2 A\)
  • \(1 + \cot^2 A = \csc^2 A\)
• Relationship between tan A and cos 2A: The identity \(\cos 2A = \dfrac{1 - \tan^2 A}{1 + \tan^2 A}\) is particularly useful when only the tangent of an angle is known and the cosine of the double angle is required.