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Trigonometric Ratio for Compound Angles (Some more Result) is considered one the most difficult concept.
Trigonometric Ratio for Compound Angles (Part 1) is considered one of the most asked concept.
39 Questions around this concept.
If , then the value of
is :
If , then
is equal to?
If $\tan A=\frac{1}{\sqrt{x\left(x^2+x+1\right)}}, \tan B=\frac{\sqrt{x}}{\sqrt{x^2+x+1}}$ and $\tan C=\left(x^{-3}+x^{-2}+x^{-1}\right)^{1 / 2}, 0<A, B, C<\frac{\pi}{2}$, then $A+B$ is equal to:
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The value of is?
If , then the value of
is?
What is the value of ?
If then the value of
is?
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Trigonometric Ratio for Compound Angles (Part 1)
The sum or difference of two or more angles is called a compound angle. If A, B and C are any angle then A + B, A - B, A + B + C, A + B - C etc all are compound angles.
1. cos (α - β) = cos α cos β + sin α sin β
2. cos (α+ β) = cos α cos β - sin α sin β
3. sin (α - β) = sin α cos β - cos α sin β
4. sin (α + β) = sin α cos β + cos α sin β
Proof:
1. cos (α - β)
Let’s consider two points on the unit circle. Point P is at an angle α from the positive x-axis with coordinates (cos α, sin α) and point Q is at an angle of β from the positive x-axis with coordinates (cos β, sin β). Note the measure of angle POQ is α − β. Label two more points: A at an angle of (α − β) from the positive x-axis with coordinates (cos(α − β), sin(α − β)); and point B with coordinates (1, 0). Triangle POQ is a rotation of triangle AOB and thus the distance from P to Q is the same as the distance from A to B.
We can find the distance from P to Q using the distance formula.
Similarly, using the distance formula we can find the distance from A to B.
Because the two distances are the same, we set them equal to each other and simplify
Finally we subtract 2 from both sides and divide both sides by −2
2. cos (α + β)
3. Sine Compound Angle Formulae
Trigonometric Ratio for Compound Angles (Part 2)
Proof:
Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying,
[Divide the numerator and denominator by cosα cos β]
For the difference of tangent, put - β in place of β in the above result
Cot Formulae
For the difference of cotangent, put - β in place of β in the above result
Trigonometric Ratio for Compound Angles (Part 3)
Proof Cotangent of the Sum and Difference of two Angles
For the difference of cotangent, put - β in place of β in the above derivation.
Trigonometric Ratio for Compound Angles (Some more Results)
Proof:
1.
2.
3.
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