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Chapter 9: Problem 80

Verify the identity: $$ \sin 2 x+1=(\sin x+\cos x)^{2} $$ (Section \(6.3, \text { Examples } 3 \text { and } 6)\)

Short Answer

Expert verified
After applying the double angle identity on the LHS and expanding the binomial on the RHS, the simplified LHS (\(2\sin(x)\cos(x) + 1\)) and RHS (\(1 + 2\sin(x)\cos(x)\)) are identical. Therefore, the given identity \(\sin 2 x+1=(\sin x+\cos x)^{2}\) is verified.

Step by step solution

01

Recall trigonometric identities

This problem involves dealing with trigonometric functions, therefore it is critical to recognize that \(\sin(2x) = 2\sin(x)\cos(x)\). This will be useful in the further steps.
02

Simplify the LHS

Let's start by simplifying the LHS. Replace \(\sin(2x)\) with \(2\sin(x)\cos(x)\) to get \(2\sin(x)\cos(x) + 1\).
03

Expand the RHS

Next, let's simplify the RHS by expanding the square of the binomial \((\sin(x) + \cos(x))^{2}\). The result is \(\sin^2(x) + 2\sin(x)\cos(x) + \cos^2(x)\). Since \(\sin^2(x) + \cos^2(x)\) is equal to 1 using the Pythagorean identity, the RHS can be simplified as \(1 + 2\sin(x)\cos(x)\).
04

Compare the simplified LHS and RHS

At this point, it's evident that the simplified LHS, \(2\sin(x)\cos(x) + 1\), is equal to the simplified RHS, \(1 + 2\sin(x)\cos(x)\). Hence, the given identity is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pythagorean identity
The Pythagorean identity is one of the fundamental relationships in trigonometry. It states:\[ \sin^2(x) + \cos^2(x) = 1 \]
This equation is pivotal because it holds true for any angle \(x\). It's derived from the Pythagorean Theorem and helps establish the relationship between sine and cosine for a triangle on the unit circle.
  • Consider a right triangle with sides \(a\), \(b\), and hypotenuse \(c\). The Pythagorean Theorem states that \(a^2 + b^2 = c^2\).
  • For the unit circle, \(c = 1\), so substituting gives \(\sin^2(x) + \cos^2(x) = 1\).
  • It's useful for simplifying expressions where sine and cosine appear together. Like in the example, we can replace \(\sin^2(x) + \cos^2(x)\) with 1.
By using this identity, you can simplify complex trigonometric expressions and verify equations more efficiently.
Double angle formula
The double angle formula is a crucial tool in trigonometry used to simplify expressions involving angles twice the measure of a given angle. For sine, the formula is:\[ \sin(2x) = 2\sin(x)\cos(x) \]
This formula helps in expressing \(\sin(2x)\) using the simpler terms \(\sin(x)\) and \(\cos(x)\). It allows for easier manipulation and verification of trigonometric identities and equations.
  • It's derived from the sum formula for sine, which states that \(\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)\).
  • By setting \(a = b = x\), you obtain \(\sin(2x) = \sin(x)\cos(x) + \cos(x)\sin(x) = 2\sin(x)\cos(x)\).
  • This simplification is especially useful when solving or verifying identities, as it breaks down more complicated functions into manageable parts.
Using the double angle formula in the exercise simplifies the left-hand side, making it possible to equate it with the expanded right-hand side.
Expanding binomials
Expanding binomials is a technique in algebra that involves multiplying two binomial expressions. It's particularly useful when working with polynomial expressions and verifying identities. To expand \((a + b)^2\), you use the distributive property, leading to:\[ (a + b)^2 = a^2 + 2ab + b^2 \]
In the context of trigonometric identities, expanding allows us to express our equations in simpler terms.
  • For the identity \((\sin x + \cos x)^2\), expanding results in \(\sin^2 x + 2\sin x \cos x + \cos^2 x\).
  • Thanks to the Pythagorean identity, \(\sin^2 x + \cos^2 x = 1\), further simplification is possible.
  • This method is essential to downloading obstacles in expressions, making problem solving more efficient and straightforward.
In the provided exercise, expanding \((\sin x + \cos x)^2\) is key to transforming the right-hand side, so it matches the left-hand side.

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Most popular questions from this chapter

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rrr} {3} & {1} & {1} \\ {-1} & {2} & {5} \end{array}\right], \quad B=\left[\begin{array}{rrr} {2} & {-3} & {6} \\ {-3} & {1} & {-4} \end{array}\right] $$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{c} {w+x+y+z=5} \\ {w+2 x-y-2 z=-1} \\ {w-3 x-3 y-z=-1} \\ {2 w-x+2 y-z=-2} \end{array}\right. $$

Will help you prepare for the material covered in the next section. Multiply: $$ \left[\begin{array}{ll} {a_{11}} & {a_{12}} \\ {a_{21}} & {a_{22}} \end{array}\right]\left[\begin{array}{ll} {1} & {0} \\ {0} & {1} \end{array}\right] $$ After performing the multiplication, describe what happens to the elements in the first matrix.

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rrr} {6} & {-3} & {5} \\ {6} & {0} & {-2} \\ {-4} & {2} & {-1} \end{array}\right], \quad B=\left[\begin{array}{rrr} {-3} & {5} & {1} \\ {-1} & {2} & {-6} \\ {2} & {0} & {4} \end{array}\right] $$

Write a system of linear equations in three or four variables to solve. Then use matrices to solve the system. A furniture company produces three types of desks: a children's model, an office model, and a deluxe model. Each desk is manufactured in three stages: cutting, construction, and finishing. The time requirements for each model and manufacturing stage are given in the following table. $$ \begin{array}{ccc} {} & {\text { Children's }} & {\text { Office }} & {\text { Deluxe }} \\\& {\text { Model }} & {\text { Model }} & {\text { Model }} \\ {\text { Cutting }} & {2 \text { hr }} & {3 \text { hr }} & {2 \text { hr }} \\\ {\text { Construction }} & {2 \text { hr }} & {1 \text { hr }} & {3 \text { hr }} \\ {\text { Finishing }} & {1 \text { hr }} & {1 \text { hr }} & {2 \text { hr }} \end{array} $$ Each week the company has available a maximum of 100 hours for cutting, 100 hours for construction, and 65 hours for finishing. If all available time must be used, how many of each type of desk should be produced each week?
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