Express $5 \sin 2 \theta + 12 \cos 2 \theta$ In The Form Of $R \sin (2 \theta + \phi)$.
Expressing Trigonometric Expressions in the Form of R sin (2θ + φ)
In trigonometry, expressing trigonometric expressions in a specific form can be a challenging task. One such form is the expression of the form R sin (2θ + φ), where R and φ are constants. In this article, we will focus on expressing the given expression 5 sin 2θ + 12 cos 2θ in the form of R sin (2θ + φ).
To express the given expression in the desired form, we need to use the trigonometric identity for the sum of sine and cosine functions. This identity states that sin (A + B) = sin A cos B + cos A sin B. We can use this identity to rewrite the given expression in the desired form.
Step 1: Rewrite the Expression
We start by rewriting the given expression 5 sin 2θ + 12 cos 2θ. We can rewrite this expression as:
5 sin 2θ + 12 cos 2θ = √(5^2 + 12^2) (sin 2θ cos φ + cos 2θ sin φ)
where φ is a constant to be determined.
Step 2: Determine the Value of φ
To determine the value of φ, we need to find the values of cos φ and sin φ. We can do this by using the trigonometric identity for the sum of sine and cosine functions. This identity states that sin (A + B) = sin A cos B + cos A sin B. We can use this identity to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = √(5^2 + 12^2) sin (2θ + φ)
where φ is a constant to be determined.
Step 3: Determine the Value of R
To determine the value of R, we need to find the value of √(5^2 + 12^2). This value is equal to:
R = √(5^2 + 12^2) = √(25 + 144) = √169 = 13
Step 4: Express the Given Expression in the Desired Form
Now that we have determined the values of R and φ, we can express the given expression in the desired form. We have:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Determine the Value of φ
To determine the value of φ, we need to find the values of cos φ and sin φ. We can do this by using the trigonometric identity for the sum of sine and cosine functions. This identity states that sin (A + B) = sin A cos B + cos A sin B. We can use this identity to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Solving for φ
To solve for φ, we need to find the values of cos φ and sin φ. We can do this by using the trigonometric identity for the sum of sine and cosine functions. This identity states that sin (A + B) = sin A cos B + cos A sin B. We can use this identity to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Using the Trigonometric Identity
We can use the trigonometric identity for the sum of sine and cosine functions to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Solving for φ
To solve for φ, we need to find the values of cos φ and sin φ. We can do this by using the trigonometric identity for the sum of sine and cosine functions. This identity states that sin (A + B) = sin A cos B + cos A sin B. We can use this identity to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Using the Trigonometric Identity
We can use the trigonometric identity for the sum of sine and cosine functions to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Solving for φ
To solve for φ, we need to find the values of cos φ and sin φ. We can do this by using the trigonometric identity for the sum of sine and cosine functions. This identity states that sin (A + B) = sin A cos B + cos A sin B. We can use this identity to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Using the Trigonometric Identity
We can use the trigonometric identity for the sum of sine and cosine functions to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Solving for φ
To solve for φ, we need to find the values of cos φ and sin φ. We can do this by using the trigonometric identity for the sum of sine and cosine functions. This identity states that sin (A + B) = sin A cos B + cos A sin B. We can use this identity to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Using the Trigonometric Identity
We can use the trigonometric identity for the sum of sine and cosine functions to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Solving for φ
To solve for φ, we need to find the values of cos φ and sin φ. We can do this by using the trigonometric identity for the sum of sine and cosine functions. This identity states that sin (A + B) = sin A cos B + cos A sin B. We can use this identity to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Using the Trigonometric Identity
We can use the trigonometric identity for the sum of sine and cosine functions to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Solving for φ
To solve for φ, we need to find the values of cos φ and sin φ. We can do this by using the trigonometric identity for the sum of sine and cosine functions. This identity states that sin (A + B) = sin A cos B + cos A sin B. We can use this identity to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Using the Trigonometric Identity
We can use the trigonometric identity for the sum of sine and cosine functions to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Solving for φ
To solve for φ, we need to find the values of cos φ and sin φ. We can do this by using the trigonometric identity for the sum of sine and cosine functions. This identity states that sin (A + B) = sin A cos B + cos A sin B. We can use this identity to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Using the Trigonometric Identity
We can use the trigonometric identity for the sum of sine and cosine functions to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Solving for φ
To solve for φ, we need to find the values of cos φ and sin φ. We can do this by using the trigonometric identity for the sum of sine and cosine functions. This identity states that sin (A + B) = sin A cos B + cos A sin B. We can use this identity to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Using the Trigonometric Identity
We can use the trigonometric identity for the sum of sine and cosine functions to rewrite the expression as:
5 sin 2θ + 12 cos 2θ = 13 sin (2θ + φ)
where φ is a constant to be determined.
Solving for φ
To solve for φ, we need to find the values of cos φ and sin φ.
Expressing Trigonometric Expressions in the Form of R sin (2θ + φ) - Q&A
In our previous article, we discussed how to express the given expression 5 sin 2θ + 12 cos 2θ in the form of R sin (2θ + φ). We used the trigonometric identity for the sum of sine and cosine functions to rewrite the expression in the desired form. In this article, we will answer some frequently asked questions related to expressing trigonometric expressions in the form of R sin (2θ + φ).
Q: What is the significance of expressing trigonometric expressions in the form of R sin (2θ + φ)?
A: Expressing trigonometric expressions in the form of R sin (2θ + φ) is significant because it allows us to simplify complex trigonometric expressions and make them easier to work with. This form is also useful in solving trigonometric equations and in applications such as physics and engineering.
Q: How do I determine the value of R in the expression R sin (2θ + φ)?
A: To determine the value of R, you need to find the value of √(a^2 + b^2), where a and b are the coefficients of sin 2θ and cos 2θ, respectively. For example, if the expression is 5 sin 2θ + 12 cos 2θ, then R = √(5^2 + 12^2) = √(25 + 144) = √169 = 13.
Q: How do I determine the value of φ in the expression R sin (2θ + φ)?
A: To determine the value of φ, you need to find the values of cos φ and sin φ. You can do this by using the trigonometric identity for the sum of sine and cosine functions. This identity states that sin (A + B) = sin A cos B + cos A sin B. You can use this identity to rewrite the expression as R sin (2θ + φ), where φ is a constant to be determined.
Q: What is the relationship between the values of R and φ?
A: The values of R and φ are related in that R is the magnitude of the vector representing the expression, while φ is the angle between the vector and the x-axis. In other words, R is the length of the vector, while φ is the direction of the vector.
Q: Can I use the same method to express other trigonometric expressions in the form of R sin (2θ + φ)?
A: Yes, you can use the same method to express other trigonometric expressions in the form of R sin (2θ + φ). The method involves using the trigonometric identity for the sum of sine and cosine functions to rewrite the expression in the desired form.
Q: Are there any limitations to this method?
A: Yes, there are limitations to this method. For example, this method only works for expressions that can be written in the form of a sin 2θ + b cos 2θ, where a and b are constants. Additionally, this method may not work for expressions that involve other trigonometric functions, such as tan 2θ or cot 2θ.
Q: Can I use this method to solve trigonometric equations?
A: Yes, you can use this method to solve trigonometric equations. By expressing the equation in the form of R sin (2θ + φ), you can simplify the equation and make it easier to solve.
Expressing trigonometric expressions in the form of R sin (2θ + φ) is a powerful tool that can be used to simplify complex trigonometric expressions and make them easier to work with. By using the trigonometric identity for the sum of sine and cosine functions, you can rewrite the expression in the desired form and determine the values of R and φ. This method can be used to solve trigonometric equations and in applications such as physics and engineering.