Solving Trigonometry with Complementary Angles

The question asks us to find the value of the expression \( 2\sqrt{3} \tan \theta + 1 \) given that \( \tan(90^\circ - \theta) = \frac{2}{\sqrt{3}} \). This problem involves trigonometric ratios and the concept of complementary angles.

Understanding Complementary Angle Identity

A key trigonometric identity states that the tangent of a complementary angle (90 degrees minus an angle) is equal to the cotangent of that angle. Mathematically, this is written as:

\( \tan(90^\circ - \theta) = \cot \theta \)

Finding the value of cot \(\theta\)

We are given that \( \tan(90^\circ - \theta) = \frac{2}{\sqrt{3}} \). Using the identity above, we can directly say:

\( \cot \theta = \frac{2}{\sqrt{3}} \)

Relating cot \(\theta\) and tan \(\theta\)

The tangent and cotangent functions are reciprocals of each other. This means:

\( \tan \theta = \frac{1}{\cot \theta} \)

Now we can find the value of \( \tan \theta \):

\( \tan \theta = \frac{1}{\frac{2}{\sqrt{3}}} \)

\( \tan \theta = \frac{\sqrt{3}}{2} \)

Evaluating the Expression

We need to find the value of \( 2\sqrt{3} \tan \theta + 1 \). We now have the value of \( \tan \theta \), which is \( \frac{\sqrt{3}}{2} \). Substitute this value into the expression:

\( 2\sqrt3 \left( \frac{\sqrt{3}}{2} \right) + 1 \)

Step-by-Step Calculation

Let's perform the multiplication:

• First, multiply \( 2\sqrt3 \) by \( \frac{\sqrt{3}}{2} \):
• \( 2\sqrt3 \times \frac{\sqrt{3}}{2} = \frac{2 \times \sqrt3 \times \sqrt3}{2} \)
• Since \( \sqrt3 \times \sqrt3 = 3 \), this becomes:
• \( \frac{2 \times 3}{2} = \frac{6}{2} = 3 \)
• Now, substitute this result back into the original expression:
• \( 3 + 1 \)
• Finally, add the numbers:
• \( 3 + 1 = 4 \)

Thus, the value of \( 2\sqrt3 \tan \theta + 1 \) is 4.

Revision Table: Key Trigonometry Concepts

Additional Information: Trigonometric Ratios

Trigonometric ratios relate the angles of a right-angled triangle to the ratios of the lengths of its sides. The basic ratios are sine (sin), cosine (cos), and tangent (tan). Their reciprocals are cosecant (csc), secant (sec), and cotangent (cot), respectively.

• \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
• \( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
• \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)
• \( \csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite}} \)
• \( \sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}} \)
• \( \cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adjacent}}{\text{Opposite}} \)

Complementary angle identities are useful for simplifying expressions and solving equations involving trigonometric functions of angles related to 90 degrees.