csc of 45
Understand the Problem
The question is asking for the value of the cosecant function (csc) for an angle of 45 degrees. To solve this, we can use the fact that csc is the reciprocal of the sine function. Since sin(45 degrees) is known to be √2/2, csc(45 degrees) will be 1/(√2/2) which simplifies to √2.
Answer
The value of $\csc(45^\circ)$ is $\sqrt{2}$.
Answer for screen readers
The value of $\csc(45^\circ)$ is $\sqrt{2}$.
Steps to Solve
- Identify the sine function value at 45 degrees
We start with the known value of the sine function. The sine of 45 degrees is given by:
$$ \sin(45^\circ) = \frac{\sqrt{2}}{2} $$
- Understand the cosecant function
The cosecant function is the reciprocal of the sine function. Therefore, we can express this relationship as:
$$ \csc(\theta) = \frac{1}{\sin(\theta)} $$
- Calculate the cosecant of 45 degrees
Now we can find the cosecant of 45 degrees by substituting the sine value we found:
$$ \csc(45^\circ) = \frac{1}{\sin(45^\circ)} = \frac{1}{\frac{\sqrt{2}}{2}} $$
- Simplify the expression
To simplify this fraction, we multiply by the reciprocal:
$$ \csc(45^\circ) = \frac{1 \times 2}{\sqrt{2}} = \frac{2}{\sqrt{2}} $$
- Rationalize the denominator
Next, we rationalize the denominator by multiplying the numerator and the denominator by $\sqrt{2}$:
$$ \csc(45^\circ) = \frac{2\sqrt{2}}{2} = \sqrt{2} $$
The value of $\csc(45^\circ)$ is $\sqrt{2}$.
More Information
The cosecant function is often used in trigonometry and is particularly useful in right triangle problems. Since $\csc(45^\circ) = \sqrt{2}$, this indicates that in a right triangle with a 45-degree angle, the ratio of the hypotenuse to the opposite side is $\sqrt{2}$ times the length of the opposite side.
Tips
- Incorrectly calculating the sine value: Sometimes people forget the specific values of sine for common angles. It's important to remember that $\sin(45^\circ) = \frac{\sqrt{2}}{2}$.
- Not using the reciprocal correctly: Make sure to apply the reciprocal relationship properly when finding cosecant.
