If 2cosθ = √3, then what is the value of tan 2θ?
√3
Solving the Trigonometry Problem: Finding tan 2θ
We are given a trigonometric equation relating cos θ and asked to find the value of tan 2θ. To solve this, we first need to find the value of θ using the given equation, then calculate 2θ, and finally find the tangent of that angle.
Step 1: Find the value of cos θ
The given equation is:
\( 2\cos\theta = \sqrt{3} \)
To isolate \(\cos\theta\), we divide both sides of the equation by 2:
\( \cos\theta = \frac{\sqrt{3}}{2} \)
Step 2: Determine the value of θ
Now we need to find the angle \(\theta\) whose cosine is \( \frac{\sqrt{3}}{2} \). We know from standard trigonometric values that the cosine of 30 degrees is \( \frac{\sqrt{3}}{2} \).
So, \( \theta = 30^\circ \).
For reference, here are some standard trigonometric values:
| Angle (\(\theta\)) | sin \(\theta\) | cos \(\theta\) | tan \(\theta\) |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | \( \frac{1}{2} \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{\sqrt{3}} \) |
| 45° | \( \frac{1}{\sqrt{2}} \) | \( \frac{1}{\sqrt{2}} \) | 1 |
| 60° | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{2} \) | \( \sqrt{3} \) |
| 90° | 1 | 0 | Undefined |
Step 3: Calculate the value of 2θ
Since \( \theta = 30^\circ \), we can find \( 2\theta \) by multiplying \(\theta\) by 2:
\( 2\theta = 2 \times 30^\circ = 60^\circ \)
Step 4: Find the value of tan 2θ
Finally, we need to find the value of tan 2θ. Substituting the value of \( 2\theta \) we just calculated:
\( \tan 2\theta = \tan 60^\circ \)
Referring to the standard trigonometric values table or recalling the value of \(\tan 60^\circ\), we know that:
\( \tan 60^\circ = \sqrt{3} \)
Therefore, the value of tan 2θ is \( \sqrt{3} \).
Revision Table: Key Trigonometric Concepts
| Concept | Description | Example |
|---|---|---|
| Cosine (cos) | Ratio of the adjacent side to the hypotenuse in a right triangle. | \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) |
| Tangent (tan) | Ratio of the opposite side to the adjacent side in a right triangle. Also \( \tan\theta = \frac{\sin\theta}{\cos\theta} \). | \( \tan 60^\circ = \frac{\sin 60^\circ}{\cos 60^\circ} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3} \) |
| Standard Angles | Common angles (0°, 30°, 45°, 60°, 90°) whose trigonometric ratios are frequently used. | Values like \( \sin 30^\circ = 1/2 \), \( \cos 45^\circ = 1/\sqrt{2} \), \( \tan 60^\circ = \sqrt{3} \). |
Additional Information on Trigonometric Functions
Trigonometric functions like sine, cosine, and tangent are fundamental in mathematics, particularly in geometry, physics, and engineering. They describe the relationships between the angles and sides of triangles.
- The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan).
- They are defined for acute angles in a right-angled triangle as ratios of its sides.
- These functions can be extended to any angle using the unit circle concept.
- Specific angle values, often called "standard angles" (like 30°, 45°, 60°), have exact trigonometric ratios that are important to remember for solving problems quickly.
- There are also reciprocal trigonometric functions: cosecant (csc), secant (sec), and cotangent (cot).
- Trigonometric identities are equations involving trigonometric functions that are true for every single value of the variable. An example is \( \sin^2\theta + \cos^2\theta = 1 \).
Understanding these basic concepts and the standard angle values is crucial for solving trigonometric problems effectively.
Important Questions from Geometry
The angles of a cyclic quadrilateral, taken in order, are x°, (3x - 30)°, (y + 30)°, and (2x - y)°. Find the measure of the smallest angle of the quadrilateral.
Length of three sides of a triangular field are 15m, 19m, and 22m respectively. What is the area of the field? (correct to one decimal place)
A triangle with vertices (3,1), (-1,0), (2,5) is:
If cos (x−y) = √3/2 and sin (x + y) = 1, where x > y, then the value of y is:
The area (in sq units) of the triangle by joining the points (3,0), (0,6) and (-5,0) is
