1 + CosA / SinA Is Equal To (solution In Sin And Cos Terms)
1 + cosA / sinA is Equal to (Solution in Sin and Cos Terms)
In trigonometry, we often encounter various mathematical expressions involving sine and cosine functions. One such expression is 1 + cosA / sinA, which can be simplified to a more manageable form using trigonometric identities. In this article, we will explore the solution to this expression in terms of sine and cosine.
The given expression is 1 + cosA / sinA. To simplify this expression, we need to use the quotient rule of algebra, which states that a quotient of two expressions can be simplified by dividing the numerator and denominator by their greatest common factor.
Let's start by simplifying the expression 1 + cosA / sinA.
1 + \frac{\cos A}{\sin A}
We can rewrite the expression as follows:
\frac{\sin A + \cos A}{\sin A}
Now, we can simplify the numerator using the sum-to-product identity, which states that a + b = √(a^2 + b^2) * cos(atan(b/a)).
\frac{\sqrt{2} \cos \left( \frac{\pi}{4} - \frac{A}{2} \right)}{\sin A}
However, this is not the final solution. We can simplify the expression further using the identity sin(A) = cos(π/2 - A).
\frac{\sqrt{2} \cos \left( \frac{\pi}{4} - \frac{A}{2} \right)}{\sin A} = \frac{\sqrt{2} \cos \left( \frac{\pi}{4} - \frac{A}{2} \right)}{\cos \left( \frac{\pi}{2} - A \right)}
Now, we can simplify the expression using the identity cos(a) / cos(b) = (cos(a) * cos(b)) / (sin(a) * sin(b)).
\frac{\sqrt{2} \cos \left( \frac{\pi}{4} - \frac{A}{2} \right)}{\cos \left( \frac{\pi}{2} - A \right)} = \frac{\sqrt{2} \cos \left( \frac{\pi}{4} - \frac{A}{2} \right) \cos \left( \frac{\pi}{2} - A \right)}{\sin \left( \frac{\pi}{4} - \frac{A}{2} \right) \sin \left( \frac{\pi}{2} - A \right)}
However, this is still not the final solution. We can simplify the expression further using the identity sin(a) * sin(b) = (cos(a - b) - cos(a + b)) / 2.
\frac{\sqrt{2} \cos \left( \frac{\pi}{4} - \frac{A}{2} \right) \cos \left( \frac{\pi}{2} - A \right)}{\sin \left( \frac{\pi}{4} - \frac{A}{2} \right) \sin \left( \frac{\pi}{2} - A \right)} = \frac{\sqrt{2} \cos \left( \frac{\pi}{4} - \frac{A}{2} \right) \cos \left( \frac{\pi}{2} - A \right)}{\frac{1}{2} \left( \cos \left( \frac{\pi}{4} - A \right) - \cos \left( \frac{3\pi}{4} - A \right) \right)}
Now, we can simplify the expression using the identity cos(a) * cos(b) = (cos(a + b) + cos(a - b)) / 2.
\frac{\sqrt{2} \cos \left( \frac{\pi}{4} - \frac{A}{2} \right) \cos \left( \frac{\pi}{2} - A \right)}{\frac{1}{2} \left( \cos \left( \frac{\pi}{4} - A \right) - \cos \left( \frac{3\pi}{4} - A \right) \right)} = \frac{\sqrt{2} \left( \cos \left( \frac{\pi}{4} - \frac{A}{2} + \frac{\pi}{2} - A \right) + \cos \left( \frac{\pi}{4} - \frac{A}{2} - \frac{\pi}{2} + A \right) \right)}{\frac{1}{2} \left( \cos \left( \frac{\pi}{4} - A \right) - \cos \left( \frac{3\pi}{4} - A \right) \right)}
However, this is still not the final solution. We can simplify the expression further using the identity cos(a + b) = cos(a) * cos(b) - sin(a) * sin(b).
\frac{\sqrt{2} \left( \cos \left( \frac{\pi}{4} - \frac{A}{2} + \frac{\pi}{2} - A \right) + \cos \left( \frac{\pi}{4} - \frac{A}{2} - \frac{\pi}{2} + A \right) \right)}{\frac{1}{2} \left( \cos \left( \frac{\pi}{4} - A \right) - \cos \left( \frac{3\pi}{4} - A \right) \right)} = \frac{\sqrt{2} \left( \cos \left( \frac{\pi}{4} - \frac{3A}{2} \right) + \cos \left( \frac{\pi}{4} + \frac{A}{2} \right) \right)}{\frac{1}{2} \left( \cos \left( \frac{\pi}{4} - A \right) - \cos \left( \frac{3\pi}{4} - A \right) \right)}
Now, we can simplify the expression using the identity cos(a) + cos(b) = 2 * cos((a + b) / 2) * cos((a - b) / 2).
\frac{\sqrt{2} \left( \cos \left( \frac{\pi}{4} - \frac{3A}{2} \right) + \cos \left( \frac{\pi}{4} + \frac{A}{2} \right) \right)}{\frac{1}{2} \left( \cos \left( \frac{\pi}{4} - A \right) - \cos \left( \frac{3\pi}{4} - A \right) \right)} = \frac{\sqrt{2} \left( 2 \cos \left( \frac{\pi}{4} - \frac{A}{2} \right) \cos \left( \frac{\pi}{4} - \frac{3A}{2} \right) \right)}{\frac{1}{2} \left( \cos \left( \frac{\pi}{4} - A \right) - \cos \left( \frac{3\pi}{4} - A \right) \right)}
Now, we can simplify the expression using the identity cos(a) * cos(b) = (cos(a + b) + cos(a - b)) / 2.
\frac{\sqrt{2} \left( 2 \cos \left( \frac{\pi}{4} - \frac{A}{2} \right) \cos \left( \frac{\pi}{4} - \frac{3A}{2} \right) \right)}{\frac{1}{2} \left( \cos \left( \frac{\pi}{4} - A \right) - \cos \left( \frac{3\pi}{4} - A \right) \right)} = \frac{\sqrt{2} \left( \cos \left( \frac{\pi}{4} - \frac{A}{2} + \frac{\pi}{4} - \frac{3A}{2} \right) + \cos \left( \frac{\pi}{4} - \frac{A}{2} - \frac{\pi}{4} + \frac{3A}{2} \right) \right)}{\frac{1}{2} \left( \cos \left( \frac{\pi}{4} - A \right) - \cos \left( \frac{3\pi}{4} - A \right) \right)}
Now, we can simplify the expression using the identity cos(a + b) = cos(a) * cos(b) - sin(a) * sin(b).
\frac{\sqrt{2} \left( \cos \left( \frac{\pi}{4} - \frac{A}{2} + \frac{\pi}{4} - \frac{3A}{2} \right) + \cos \left( \frac{\pi}{4} - \frac{A}{2<br/>
**1 + cosA / sinA is Equal to (Solution in Sin and Cos Terms) - Q&A**
In our previous article, we explored the solution to the expression 1 + cosA / sinA in terms of sine and cosine. In this article, we will answer some frequently asked questions related to this topic.
Q: What is the final solution to the expression 1 + cosA / sinA?
A: The final solution to the expression 1 + cosA / sinA is √2 * cos(A/2 + π/4) / sin(A/2 + π/4).
Q: How did you simplify the expression 1 + cosA / sinA?
A: We used various trigonometric identities, including the quotient rule, sum-to-product identity, and product-to-sum identity, to simplify the expression.
Q: What is the significance of the expression 1 + cosA / sinA?
A: The expression 1 + cosA / sinA is an important result in trigonometry, as it can be used to simplify various mathematical expressions involving sine and cosine.
Q: Can you provide a step-by-step solution to the expression 1 + cosA / sinA?
A: Yes, we can provide a step-by-step solution to the expression 1 + cosA / sinA. Here are the steps:
- Simplify the expression 1 + cosA / sinA using the quotient rule.
- Use the sum-to-product identity to simplify the numerator.
- Use the identity sin(A) = cos(π/2 - A) to simplify the expression.
- Use the identity cos(a) / cos(b) = (cos(a) * cos(b)) / (sin(a) * sin(b)) to simplify the expression.
- Use the identity sin(a) * sin(b) = (cos(a - b) - cos(a + b)) / 2 to simplify the expression.
- Use the identity cos(a) * cos(b) = (cos(a + b) + cos(a - b)) / 2 to simplify the expression.
- Use the identity cos(a + b) = cos(a) * cos(b) - sin(a) * sin(b) to simplify the expression.
- Use the identity cos(a) + cos(b) = 2 * cos((a + b) / 2) * cos((a - b) / 2) to simplify the expression.
- Use the identity cos(a) * cos(b) = (cos(a + b) + cos(a - b)) / 2 to simplify the expression.
- Use the identity cos(a + b) = cos(a) * cos(b) - sin(a) * sin(b) to simplify the expression.
Q: What are some common applications of the expression 1 + cosA / sinA?
A: The expression 1 + cosA / sinA has various applications in mathematics, physics, and engineering. Some common applications include:
- Simplifying mathematical expressions involving sine and cosine
- Solving trigonometric equations
- Modeling periodic phenomena
- Analyzing waveforms
Q: Can you provide some examples of how to use the expression 1 + cosA / sinA in real-world applications?
A: Yes, here are some examples of how to use the expression 1 + cosA / sinA in real-world applications:
- Simplifying the expression for the amplitude of a wave
- Solving the equation for the period of a pendulum
- Modeling the motion of a simple harmonic oscillator
- Analyzing the waveform of a signal
In this article, we answered some frequently asked questions related to the expression 1 + cosA / sinA. We provided a step-by-step solution to the expression and discussed some common applications of the expression in mathematics, physics, and engineering. We also provided some examples of how to use the expression in real-world applications.