Solving the Trigonometric Equation and Identity

The problem asks us to find the value of \(\tan(45^\circ + \theta)\), given the relationship \(3\cos\theta = 4\sin\theta\). To solve this, we first need to find the value of \(\tan\theta\) from the given condition, and then use the tangent addition formula.

Step 1: Finding the Value of tanθ

We are given the equation:

\(3\cos\theta = 4\sin\theta\)

To find \(\tan\theta\), we can rearrange this equation. Recall that \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\). We can divide both sides of the given equation by \(\cos\theta\) (assuming \(\cos\theta \neq 0\)).

\(\dfrac{3\cos\theta}{\cos\theta} = \dfrac{4\sin\theta}{\cos\theta}\)

This simplifies to:

\(3 = 4 \left(\dfrac{\sin\theta}{\cos\theta}\right)\)

\(3 = 4\tan\theta\)

Now, divide both sides by 4 to isolate \(\tan\theta\):

\(\tan\theta = \dfrac{3}{4}\)

Step 2: Using the Tangent Addition Formula

We need to find the value of \(\tan(45^\circ + \theta)\). We can use the tangent addition formula, which states:

\(\tan(A+B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}\)

In our case, \(A = 45^\circ\) and \(B = \theta\). So, the formula becomes:

\(\tan(45^\circ + \theta) = \dfrac{\tan 45^\circ + \tan \theta}{1 - \tan 45^\circ \tan \theta}\)

Step 3: Substituting Known Values and Calculating

We know that the value of \(\tan 45^\circ\) is 1. We also found that \(\tan\theta = \dfrac{3}{4}\) from Step 1.

Substitute these values into the formula:

\(\tan(45^\circ + \theta) = \dfrac{1 + \dfrac{3}{4}}{1 - 1 \cdot \dfrac{3}{4}}\)

Simplify the expression:

\(\tan(45^\circ + \theta) = \dfrac{1 + \dfrac{3}{4}}{1 - \dfrac{3}{4}}\)

To simplify the numerator and the denominator, find a common denominator (which is 4):

Numerator: \(1 + \dfrac{3}{4} = \dfrac{4}{4} + \dfrac{3}{4} = \dfrac{4+3}{4} = \dfrac{7}{4}\)

Denominator: \(1 - \dfrac{3}{4} = \dfrac{4}{4} - \dfrac{3}{4} = \dfrac{4-3}{4} = \dfrac{1}{4}\)

Now, substitute these back into the main expression:

\(\tan(45^\circ + \theta) = \dfrac{\dfrac{7}{4}}{\dfrac{1}{4}}\)

Dividing by a fraction is the same as multiplying by its reciprocal:

\(\tan(45^\circ + \theta) = \dfrac{7}{4} \times \dfrac{4}{1}\)

Cancel out the 4s:

\(\tan(45^\circ + \theta) = 7\)

Thus, the value of \(\tan(45^\circ + \theta)\) is 7.

Revision Table: Key Trigonometric Identities

Additional Information on Trigonometry

Trigonometry deals with the relationships between the sides and angles of triangles. The basic trigonometric functions are sine, cosine, and tangent. These functions have various identities and formulas that are useful for solving trigonometric equations and simplifying expressions.

• Angle Sum and Difference Formulas: Besides the tangent addition formula, there are similar formulas for sine and cosine:
  • \(\sin(A+B) = \sin A \cos B + \cos A \sin B\)
  • \(\sin(A-B) = \sin A \cos B - \cos A \sin B\)
  • \(\cos(A+B) = \cos A \cos B - \sin A \sin B\)
  • \(\cos(A-B) = \cos A \cos B + \sin A \sin B\)
  • \(\tan(A-B) = \dfrac{\tan A - \tan B}{1 + \tan A \tan B}\)
• Special Angles: It's important to remember the trigonometric values for special angles like \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), \(90^\circ\), etc.
  • \(\sin 30^\circ = \dfrac{1}{2}, \cos 30^\circ = \dfrac{\sqrt{3}}{2}, \tan 30^\circ = \dfrac{1}{\sqrt{3}}\)
  • \(\sin 45^\circ = \dfrac{1}{\sqrt{2}}, \cos 45^\circ = \dfrac{1}{\sqrt{2}}, \tan 45^\circ = 1\)
  • \(\sin 60^\circ = \dfrac{\sqrt{3}}{2}, \cos 60^\circ = \dfrac{1}{2}, \tan 60^\circ = \sqrt{3}\)
• Quadrant Rules: The sign of trigonometric functions changes depending on the quadrant the angle lies in. This is often remembered using the "All Silver Tea Cups" rule (All positive in Q1, Sine positive in Q2, Tangent positive in Q3, Cosine positive in Q4).

These concepts and formulas are fundamental to solving various problems in trigonometry, physics, engineering, and other fields.