Streamlining Inventive Solutions for SSC CGL Trigonometry Problems
Can you solve the trigonometry problem in 25 seconds? We'll show you an inventive approach that skips traditional, lengthy calculations to solve it faster.
Some problems might not appear challenging at first glance and seem quick to solve using standard methods. However, these conventional solutions often involve unnecessary steps. Here, we present both a traditional method and a faster, inventive approach utilizing an embedded useful pattern and powerful problem-solving technique.
Chosen Problem
If $ cot \theta + cosec \theta = 3 $, and $ \theta $ is an acute angle, find $ cos \theta $.
- $1$
- $\displaystyle\frac{1}{2}$
- $\displaystyle\frac{4}{5}$
- $\displaystyle\frac{3}{4}$
Conventional Approach
This approach depends primarily on using the identity $ cosec^2 \theta - cot^2 \theta = 1 $ by squaring the given equation. Thus $cot \theta$ is isolated to obtain $tan \theta$. Aim is to solve for $ sec \theta $ using the identity $ 1 + tan^2 \theta = sec^2 \theta $. It needs a second operation of squaring.
Given: $ cot \theta + cosec \theta = 3 $
Thus, $ cosec \theta = 3 - cot \theta $
Squaring both sides, we get:
$ cosec^2 \theta = 9 - 6cot \theta + cot^2 \theta $
Or, $ 9 - 6cot \theta = cosec^2 \theta - cot^ \theta = 1 $
Or, $ 6cot \theta = 8 $
Or, $ tan \theta =\displaystyle\frac{3}{4} $.
Squaring again and adding 1:
$ tan^2 \theta + 1 = sec^2 \theta = \displaystyle\frac{25}{16} $
As $ \theta $ is acute, $ sec \theta $ is positive:
$ sec \theta = \displaystyle\frac{5}{4} $
Therefore, $ cos \theta = \displaystyle\frac{4}{5} $.
Answer: Option c: $ \displaystyle\frac{4}{5} $.
Inventive Solution: Efficient Problem-Solving
Using the principle of friendly trigonometric function pairs, we identify a faster route to the solution. The key is recognizing useful patterns and simplifying expressions early.
The friendly trigonometric function pair:
In the well-known identity $cosec^2 \theta - cot^2 \theta = 1$, the LHS is broken up into two factors and a powerful relation follows:
$cosec \theta + cot \theta = \displaystyle\frac{1}{cosec \theta - cot \theta}$
This way, the two functions $cosec \theta$ and $cot \theta$ form a friendly trigonometric function pair highly useful in simplifying and solving trigonometric problems elegantly and quickly. This is inventive trigonometric problem solving.
Given expression: $ cot \theta + cosec \theta = 3 $
Using the principle:
$\displaystyle\frac{1}{cosec \theta - cot \theta} = 3 $.
Thus, $ cosec \theta - cot \theta = \displaystyle\frac{1}{3} $.
Adding the two equations eliminates $ cot \theta $, simplifying to:
$ 2cosec \theta = 3 + \displaystyle\frac{1}{3} = \displaystyle\frac{10}{3} $
So, $ cosec \theta = \displaystyle\frac{5}{3} $, and therefore:
$ sin \theta = \displaystyle\frac{3}{5} $
$ cos \theta = \sqrt{1 - \left(\displaystyle\frac{3}{5}\right)^2} =\displaystyle \frac{4}{5} $, as $\theta$ is an acute angle.
Answer: Option c: $\displaystyle \frac{4}{5} $.
Solved in a minimum number of six steps using only linear first order trigonometric functions, bypassing the need of squaring twice and saving as many as four steps.
Conclusion
This refined approach not only simplifies the problem but also demonstrates the power of understanding and applying trigonometric identities effectively. By focusing on linear expressions and minimizing the order of terms, the problem-solving process is streamlined, achieving solutions more efficiently faster.
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