Math Problem Statement

If 7tan(α) = 1 and √(10 - 10cos(2β)) = √2, with α, β ∈ (0, π/2), then the value of tan(α) + tan(2β) is:

Solution

Let's break down and solve the question visible in the image.

Given:

  1. 7tan(α)=17 \tan(\alpha) = 1, so tan(α)=17\tan(\alpha) = \frac{1}{7}
  2. 1010cos(2β)=2\sqrt{10 - 10 \cos(2\beta)} = \sqrt{2}

We are asked to find the value of tan(α)+tan(2β)\tan(\alpha) + \tan(2\beta), where α,β(0,π2)\alpha, \beta \in \left( 0, \frac{\pi}{2} \right).

Step 1: Solve for tan(2β)\tan(2\beta)

We are given: 1010cos(2β)=2\sqrt{10 - 10\cos(2\beta)} = \sqrt{2} Squaring both sides: 1010cos(2β)=210 - 10 \cos(2\beta) = 2 10cos(2β)=810 \cos(2\beta) = 8 cos(2β)=45\cos(2\beta) = \frac{4}{5} Now, using the identity: tan(2β)=sin(2β)cos(2β)\tan(2\beta) = \frac{\sin(2\beta)}{\cos(2\beta)} We know: sin2(2β)+cos2(2β)=1\sin^2(2\beta) + \cos^2(2\beta) = 1 sin2(2β)=1(45)2=11625=925\sin^2(2\beta) = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} Thus: sin(2β)=35\sin(2\beta) = \frac{3}{5} Now: tan(2β)=3545=34\tan(2\beta) = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}

Step 2: Calculate tan(α)+tan(2β)\tan(\alpha) + \tan(2\beta)

From earlier, we know: tan(α)=17,tan(2β)=34\tan(\alpha) = \frac{1}{7}, \quad \tan(2\beta) = \frac{3}{4} Adding these: tan(α)+tan(2β)=17+34\tan(\alpha) + \tan(2\beta) = \frac{1}{7} + \frac{3}{4} Finding a common denominator: tan(α)+tan(2β)=4+2128=2528\tan(\alpha) + \tan(2\beta) = \frac{4 + 21}{28} = \frac{25}{28}

Thus, the value of tan(α)+tan(2β)\tan(\alpha) + \tan(2\beta) is 2528\frac{25}{28}.

Answer:

The correct option is (1) 2528\frac{25}{28}.

Would you like more details or have any questions?

Here are five related questions for practice:

  1. What is the double angle identity for tan(2β)\tan(2\beta)?
  2. How do you derive the value of tan(α)\tan(\alpha) if 7tan(α)=17\tan(\alpha) = 1?
  3. What is the value of cos(2β)\cos(2\beta) when 1010cos(2β)=2\sqrt{10 - 10 \cos(2\beta)} = \sqrt{2}?
  4. How do you calculate tan(α)+tan(2β)\tan(\alpha) + \tan(2\beta) if α\alpha and β\beta are given in terms of radians?
  5. What is the sum of angles formula for tangent?

Tip: Always use trigonometric identities to simplify expressions before plugging in values.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Algebra
Double Angle Identities

Formulas

tan(2β) = 2tan(β) / (1 - tan²(β))
cos(2β) = 2cos²(β) - 1
tan(α) + tan(β) = (tan(α) + tan(β)) / (1 - tan(α)tan(β))

Theorems

Pythagorean Identity
Double Angle Identity

Suitable Grade Level

Grades 11-12