Problem 63
Question
Solve the following equations and tick the correct one. The general solution of the equation \(\sin 100 x-\cos 100 x=1\), is (a) \(2 n \pi+\frac{\pi}{3}, n \in z\) (b) \(n \pi+\frac{\pi}{2}, n \in z\) (c) \(n \pi+\frac{\pi}{4} \pi, n \in z\) (d) \(2 n \pi \frac{\pi}{3}, n \in z\)
Step-by-Step Solution
Verified Answer
The correct solution is \(x = n \pi + \frac{\pi}{2}\), which corresponds to option (b).
1Step 1: Stating the Equation
The problem is to solve the equation \( \sin 100 x - \cos 100x = 1\).
2Step 2: Rewriting the Equation
The left hand side of the equation is in similar form to the trigonometric identity \( \sin(\frac{\pi}{4} - x) = \cos x - \sin x\). So let's rewrite the equation as \( \sqrt{2} \sin(100x - \frac{\pi}{4}) = 1\).
3Step 3: Calculating x
To isolate x, take inverse sine on both sides to get \( 100x - \frac{\pi}{4}) = \frac{\pi}{4} + n\pi \). Then, solve for x to obtain \( x = \frac{1}{100} (\frac{\pi}{2} + n\pi) \)
4Step 4: Comparing the Results
By comparing the solution with the given options, it is clear that it is equal to option b. Hence, \( x = n \pi + \frac{\pi}{2}\), where \( n \in z \).
Key Concepts
Solving Trigonometric EquationsTrigonometric IdentitiesInverse Trigonometric Functions
Solving Trigonometric Equations
When it comes to solving trigonometric equations, the goal is to find all the angles (or values of the variable) that make the equation true. These problems can often be solved by applying various algebraic and trigonometric techniques.
For example, see the equation in our exercise: \( \sin(100x) - \cos(100x) = 1 \). The strategy involves expressing all the trigonometric functions in terms of a single function and then isolating the variable.
Working with this type of equation requires familiarity with trigonometric identities, which can help us to simplify and rewrite the expressions for easier solution. Another technique is to employ inverse trigonometric functions to find the angle which corresponds to a given trigonometric value.
In our step-by-step solution, the equation was already simplified using an appropriate identity. Then, by applying the inverse sine function, we can determine the general solution for the variable. But remember that since trigonometric functions are periodic, there will potentially be an infinite number of solutions, often expressed in terms of the variable plus a constant times any integer (notated as 'n' in mathematics), which accounts for the repeated values over intervals of the function's period.
For example, see the equation in our exercise: \( \sin(100x) - \cos(100x) = 1 \). The strategy involves expressing all the trigonometric functions in terms of a single function and then isolating the variable.
Working with this type of equation requires familiarity with trigonometric identities, which can help us to simplify and rewrite the expressions for easier solution. Another technique is to employ inverse trigonometric functions to find the angle which corresponds to a given trigonometric value.
In our step-by-step solution, the equation was already simplified using an appropriate identity. Then, by applying the inverse sine function, we can determine the general solution for the variable. But remember that since trigonometric functions are periodic, there will potentially be an infinite number of solutions, often expressed in terms of the variable plus a constant times any integer (notated as 'n' in mathematics), which accounts for the repeated values over intervals of the function's period.
Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variables involved. They play a critical role in simplifying trigonometric expressions and solving trigonometric equations. Common identities include the Pythagorean identities, quotient identities, and co-function identities among others.
In the given exercise, the identity \( \sin(\frac{\pi}{4} - x) = \cos x - \sin x \) was used to rewrite the original equation into a form that can be solved. This knowledge is not just about memorization; understanding these identities allows you to manipulate and transform equations, making complex problems more approachable.
It's important to note that selecting the right identity can sometimes make the difference between an easy solution and a complicated one. This will come with practice and experience, as you learn to see patterns and recognize which identities may be helpful in a given situation.
In the given exercise, the identity \( \sin(\frac{\pi}{4} - x) = \cos x - \sin x \) was used to rewrite the original equation into a form that can be solved. This knowledge is not just about memorization; understanding these identities allows you to manipulate and transform equations, making complex problems more approachable.
It's important to note that selecting the right identity can sometimes make the difference between an easy solution and a complicated one. This will come with practice and experience, as you learn to see patterns and recognize which identities may be helpful in a given situation.
Inverse Trigonometric Functions
Inverse trigonometric functions are the inverses of the trigonometric functions. They allow us to work backwards from a known trigonometric value to find the angle that produced it. The main functions are the arcsine (\( \sin^{-1} \)), arccosine (\( \cos^{-1} \)), and arctangent (\( \tan^{-1} \)).
Just as taking the square root is the inverse operation of squaring a number, taking the inverse trigonometric function is the inverse operation of taking a trigonometric function of an angle. However, it gets more complex, as trigonometric functions are not one-to-one unless their domain is restricted.
In our solution, the inverse sine function (or arcsine) was used to solve for \( x \). This provided us with an angle, but due to the periodic nature of sine, we have to include all possible solutions by adding \( n\pi \), where 'n' is an integer. This ensures that all solutions are accounted for, across the infinite number of cycles that a trigonometric function goes through.
Just as taking the square root is the inverse operation of squaring a number, taking the inverse trigonometric function is the inverse operation of taking a trigonometric function of an angle. However, it gets more complex, as trigonometric functions are not one-to-one unless their domain is restricted.
In our solution, the inverse sine function (or arcsine) was used to solve for \( x \). This provided us with an angle, but due to the periodic nature of sine, we have to include all possible solutions by adding \( n\pi \), where 'n' is an integer. This ensures that all solutions are accounted for, across the infinite number of cycles that a trigonometric function goes through.
Other exercises in this chapter
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