Understanding the Value of θ Given a Trigonometric Function

In the field of trigonometry, we often deal with various trigonometric functions such as sine, cosine, and tangent. When given a specific trigonometric condition, we can solve for the angle θ. This article will delve into the process of finding the value of θ when θ is an acute angle and the given condition is:

cosθ frac{1}{2} tanθ

Step 1: Simplifying the Given Condition

Let's start by simplifying the given condition. We are given:

cosθ frac{1}{2} tanθ

Recall that tanθ frac{sinθ}{cosθ}. Substituting this into our equation, we get:

cosθ frac{1}{2} frac{sinθ}{cosθ}

Multiplying both sides by cosθ to eliminate the fraction:

cos^2θ frac{1}{2} sinθ

Rewriting this in terms of cosine, we obtain:

cos^2θ frac{1}{2} sqrt{1 - cos^2θ}

Step 2: Solving for cosθ

Let cosθ t. Substituting t into our equation:

t^2 frac{1}{2} sqrt{1 - t^2}

Squaring both sides to remove the square root:

t^4 frac{1}{4}(1 - t^2)

Multiplying through by 4:

4t^4 t^2 - 1 0

This is a quadratic equation in terms of t^2. Letting u t^2 and rewriting the equation:

4u^2 u - 1 0

Step 3: Solving the Quadratic Equation

We can solve this quadratic equation using the quadratic formula:

u frac{-b pm sqrt{b^2 - 4ac}}{2a}

Here, a 4, b 1, and c -1. Substituting these values in:

u frac{-1 pm sqrt{1^2 - 4 cdot 4 cdot (-1)}}{2 cdot 4}

Which simplifies to:

u frac{-1 pm sqrt{1 16}}{8}

u frac{-1 pm sqrt{17}}{8}

We have two possible solutions for u:

u frac{-1 sqrt{17}}{8} and u frac{-1 - sqrt{17}}{8}

Since u t^2 and t represents the cosine of an angle, it must be a positive value. Therefore, we discard the negative solution:

u frac{-1 sqrt{17}}{8}

Thus, we have:

t^2 frac{-1 sqrt{17}}{8}

And:

t cosθ sqrt{frac{-1 sqrt{17}}{8}}

Conclusion

Given the condition that cosθ frac{1}{2} tanθ, we found that:

cosθ sqrt{frac{-1 sqrt{17}}{8}}

This value corresponds to the cosine of the acute angle θ. It's important to note that in trigonometry, the cosine of an angle can range from 0 to 1 for acute angles. Therefore, the solution provided is valid for the acute angle θ.

To summarize, the key steps involved were:

Simplify the given condition. Substitute and rearrange to form a quadratic equation. Solve the quadratic equation to find the value of t. Identify the valid solution for t as representing the cosine of an acute angle.

This approach can be applied to a wide range of similar problems in trigonometry, providing a systematic way to solve for angles in various trigonometric equations.