Find Two Acute Angles That Satisfy The Equation Sin ( 3 X + 9 ) = Cos ( X + 1 \sin(3x + 9) = \cos(x + 1 Sin ( 3 X + 9 ) = Cos ( X + 1 ]. Check That Your Answers Make Sense.The Smaller Angle Is □ \square □ .The Larger Angle Is □ \square □ .
Introduction
In this article, we will explore the problem of finding two acute angles that satisfy the equation sin(3x + 9) = cos(x + 1). This equation involves trigonometric functions and requires us to use various identities and properties to solve for the angles. We will start by analyzing the given equation and then proceed to solve for the angles using different methods.
Understanding the Equation
The given equation is sin(3x + 9) = cos(x + 1). To solve this equation, we need to use the identity that relates sine and cosine functions. This identity is cos(x) = sin(90 - x) for all angles x.
Using the Co-function Identity
Using the co-function identity, we can rewrite the given equation as sin(3x + 9) = sin(90 - (x + 1)). This simplifies to sin(3x + 9) = sin(89 - x).
Equating Angles
Since the sine function is periodic with a period of 360°, we can equate the angles inside the sine functions. This gives us the equation 3x + 9 = 89 - x.
Solving for x
To solve for x, we need to isolate x on one side of the equation. We can do this by adding x to both sides of the equation, which gives us 4x + 9 = 89. Then, we can subtract 9 from both sides of the equation, which gives us 4x = 80.
Finding the Value of x
To find the value of x, we need to divide both sides of the equation by 4. This gives us x = 20.
Finding the Angles
Now that we have the value of x, we can find the angles that satisfy the equation. We can substitute x = 20 into the original equation sin(3x + 9) = cos(x + 1) to get sin(3(20) + 9) = cos(20 + 1). This simplifies to sin(69) = cos(21).
Using the Co-function Identity Again
Using the co-function identity again, we can rewrite the equation sin(69) = cos(21) as sin(69) = sin(90 - 21). This simplifies to sin(69) = sin(69).
Finding the Angles
Since the sine function is periodic with a period of 360°, we can equate the angles inside the sine functions. This gives us the equation 69 = 69.
Finding the Smaller Angle
To find the smaller angle, we need to find the angle that is less than 90°. We can do this by subtracting 69 from 90, which gives us 21.
Finding the Larger Angle
To find the larger angle, we need to find the angle that is greater than 90°. We can do this by adding 69 to 90, which gives us 159.
Checking the Answers
To check our answers, we need to make sure that they satisfy the original equation sin(3x + 9) = cos(x + 1). We can do this by substituting x = 20 into the original equation to get sin(3(20) + 9) = cos(20 + 1). This simplifies to sin(69) = cos(21).
Conclusion
In this article, we found two acute angles that satisfy the equation sin(3x + 9) = cos(x + 1). The smaller angle is 21° and the larger angle is 159°. We checked our answers by substituting x = 20 into the original equation to make sure that they satisfy the equation.
Final Answer
The final answer is: and
Introduction
In our previous article, we explored the problem of finding two acute angles that satisfy the equation sin(3x + 9) = cos(x + 1). We used various identities and properties to solve for the angles and found that the smaller angle is 21° and the larger angle is 159°. In this article, we will answer some common questions related to this problem.
Q: What is the equation sin(3x + 9) = cos(x + 1) trying to solve?
A: The equation sin(3x + 9) = cos(x + 1) is trying to find two acute angles that satisfy the equation. The equation involves trigonometric functions and requires us to use various identities and properties to solve for the angles.
Q: How do we use the co-function identity to solve the equation?
A: We use the co-function identity to rewrite the equation sin(3x + 9) = cos(x + 1) as sin(3x + 9) = sin(90 - (x + 1)). This simplifies to sin(3x + 9) = sin(89 - x).
Q: How do we equate the angles inside the sine functions?
A: We equate the angles inside the sine functions by using the periodicity of the sine function. This gives us the equation 3x + 9 = 89 - x.
Q: How do we solve for x?
A: We solve for x by adding x to both sides of the equation, which gives us 4x + 9 = 89. Then, we can subtract 9 from both sides of the equation, which gives us 4x = 80. Finally, we can divide both sides of the equation by 4, which gives us x = 20.
Q: How do we find the angles that satisfy the equation?
A: We find the angles that satisfy the equation by substituting x = 20 into the original equation sin(3x + 9) = cos(x + 1). This simplifies to sin(69) = cos(21).
Q: How do we check our answers?
A: We check our answers by substituting x = 20 into the original equation sin(3x + 9) = cos(x + 1). This simplifies to sin(69) = cos(21), which is true.
Q: What are the two acute angles that satisfy the equation?
A: The two acute angles that satisfy the equation are 21° and 159°.
Q: Why are these angles acute?
A: These angles are acute because they are less than 90°.
Q: How do we know that these angles are the only solutions?
A: We know that these angles are the only solutions because we used the periodicity of the sine function to equate the angles inside the sine functions.
Q: Can we find other solutions to the equation?
A: No, we cannot find other solutions to the equation because the sine function is periodic with a period of 360°.
Conclusion
In this article, we answered some common questions related to the problem of finding two acute angles that satisfy the equation sin(3x + 9) = cos(x + 1). We used various identities and properties to solve for the angles and found that the smaller angle is 21° and the larger angle is 159°.
Final Answer
The final answer is: and