
Find using a half- angle formula.
Answer
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Hint: In this problem, we have to find the value of using a half -angle formula. We will use the half angle formula for tangent as follows.
We will put in a half angle formula and then solve by cross multiplying to get the desired value. We will also use the value of .
Complete step by step solution:
This question is based on application of trigonometric formulas. Trigonometric formula is based on the relationship between T-ratio of angles, identify sides etc.
Half angle formula is based on the formula for the sum of two angles.
For example,
If we put then the formula becomes
Considering the given question, we have to find the value of .
From half angle formula we have,
Let, Then .
Putting in above half angle formula we get
We know that
Hence we have,
On cross multiplication we have,
Adding to both sides, we have,
Let , then we have,
This is a quadratic equation. We know that the solution of quadratic equation is given by .
Here, , and .
Hence ,
Hence , .
Therefore, or
Since , lies in the first quadrant .
Therefore the value of is positive.
Hence,
Hence the value of is
So, the correct answer is “ ”.
Note: Values of all T-ratios are positive in the first quadrant. While In second quadrant, only the value of sine is positive. In the third quadrant, only tangent is positive and in the fourth quadrant, only cosine is positive.
Quadratic equations can also be solved by splitting the middle term such that sum of terms is middle term and product is constant term.
Some important trigonometry half angle formula are
We will put
Complete step by step solution:
This question is based on application of trigonometric formulas. Trigonometric formula is based on the relationship between T-ratio of angles, identify sides etc.
Half angle formula is based on the formula for the sum of two angles.
For example,
If we put
Considering the given question, we have to find the value of
From half angle formula we have,
Let,
Putting
We know that
Hence we have,
On cross multiplication we have,
Adding
Let ,
This is a quadratic equation. We know that the solution of quadratic equation
Here,
Hence ,
Hence ,
Therefore,
Since ,
Therefore the value of
Hence,
Hence the value of
So, the correct answer is “
Note: Values of all T-ratios are positive in the first quadrant. While In second quadrant, only the value of sine is positive. In the third quadrant, only tangent is positive and in the fourth quadrant, only cosine is positive.
Quadratic equations can also be solved by splitting the middle term such that sum of terms is middle term and product is constant term.
Some important trigonometry half angle formula are
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