Chapter 9: Problem 20
Simplify $$ (\sin \theta+\cos \theta)^{2}-\sin 2 \theta $$
Short Answer
Expert verified
Question: Simplify the expression \((\sin \theta+\cos \theta)^{2}-\sin 2 \theta\).
Answer: The simplified expression is \(1 - \sin\theta\cos\theta\).
Step by step solution
01
Expand the Square
First, expand the square term \((\sin \theta+\cos \theta)^{2}\). This can be done using the distributive property (FOIL method), so we get:
$$
(\sin \theta+\cos \theta)^{2} = (\sin \theta + \cos \theta)(\sin \theta + \cos \theta)
$$
$$
= (\sin \theta)(\sin \theta) + (\sin \theta)(\cos \theta) + (\cos \theta)(\sin \theta) + (\cos \theta)(\cos \theta)
$$
$$
= \sin^{2}\theta + \sin \theta \cos \theta + \cos \theta \sin \theta + \cos^{2}\theta
$$
02
Use Trigonometric Properties
Now, use the Pythagorean identity, which states that:
$$
\sin^{2}\theta + \cos^{2}\theta = 1
$$
So our expanded expression now becomes:
$$
1 + \sin \theta \cos \theta + \cos \theta \sin \theta
$$
Next, use the double angle formula for sine, which states that:
$$
\sin{2\theta} = 2\sin\theta\cos\theta
$$
So we can rewrite the given expression as:
$$
(\sin \theta+\cos \theta)^{2}-\sin 2 \theta = 1 + \sin \theta \cos \theta + \cos \theta \sin \theta - 2\sin\theta\cos\theta
$$
03
Simplify the Expression
Now, combine like terms to simplify the expression further:
$$
1 + \sin \theta \cos \theta + \cos \theta \sin \theta - 2\sin\theta\cos\theta = 1 - \sin\theta\cos\theta
$$
So the final simplified expression is:
$$
(\sin \theta+\cos \theta)^{2}-\sin 2 \theta = 1 - \sin\theta\cos\theta
$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Pythagorean Identity
The Pythagorean identity is an essential element of trigonometry that links the squares of sine and cosine functions of an angle to the number 1. It is written as \[\sin^2\theta + \cos^2\theta = 1.\] This relationship is derived from the Pythagorean theorem applied to a unit circle, where the radius is 1. It's a powerful tool because it allows us to simplify trigonometric expressions by replacing \(\sin^2\theta\) or \(\cos^2\theta\) with the complementary trigonometric function squared. For example, if you have an expression involving \(\sin^2\theta\), you can replace it with \(1 - \cos^2\theta\), and vice versa.
Consider an exercise where you must simplify the expression \(\sin\theta + \cos\theta\)^2 - \sin 2\theta.\ Using the Pythagorean identity, we know that \(\sin^2\theta + \cos^2\theta\) simplifies to 1. Therefore, part of the expression within our exercise can be immediately reduced, which is a perfect demonstration of the practical use of the Pythagorean identity.
Consider an exercise where you must simplify the expression \(\sin\theta + \cos\theta\)^2 - \sin 2\theta.\ Using the Pythagorean identity, we know that \(\sin^2\theta + \cos^2\theta\) simplifies to 1. Therefore, part of the expression within our exercise can be immediately reduced, which is a perfect demonstration of the practical use of the Pythagorean identity.
Applying the Double Angle Formula for Sine
What is the Double Angle Formula?
The double angle formula for sine, which is given by \[\sin 2\theta = 2\sin\theta\cos\theta,\] allows us to express the sine of a double angle in terms of the sine and cosine of the original angle. This formula is crucial when simplifying trigonometric expressions involving angles multiplied by two. It's obtained from the sum of angles formula for sine and is particularly helpful in transforming trigonometric functions to facilitate integration, differentiation, or simplification of equations.In our example, the original expression includes \(\sin 2\theta\), which we can now replace using the double angle formula. By doing so, the expression \(\sin\theta + \cos\theta\)^2 - \sin 2\theta\ transforms, leading to a cancellation that simplifies the overall expression — a practical application of this identity.
Relationship Between Sine and Cosine
Sine and cosine are the primary trigonometric functions that describe the relationship of the sides of a right-angled triangle to its angles, as well as the coordinates on a unit circle. The sine function \(\sin\theta\) represents the ratio of the opposite side to the hypotenuse, while the cosine function \(\cos\theta\) represents the ratio of the adjacent side to the hypotenuse. Although they are different functions, they have a unique relationship and can be derived from one another when the angle \(\theta\) completes more than a quarter of a cycle on the unit circle.
In fact, sine and cosine are often present in complementary trigonometric identities, as seen in the exercise where \(\sin\theta\cos\theta\) terms appear. These functions are not only foundational in trigonometry but also in calculus, physics, and engineering. By understanding their relationship and behavior, as evidenced in the given exercise, you can solve complex trigonometric problems with greater ease.
In fact, sine and cosine are often present in complementary trigonometric identities, as seen in the exercise where \(\sin\theta\cos\theta\) terms appear. These functions are not only foundational in trigonometry but also in calculus, physics, and engineering. By understanding their relationship and behavior, as evidenced in the given exercise, you can solve complex trigonometric problems with greater ease.