Problem 132 - Precalculus

Question

For the following exercises, prove the identity given. $$ (\sin t-\cos t)^{2}=1-\sin (2 t) $$

Step-by-Step Solution

Answer

The identity is proven: \((\sin t - \cos t)^2 = 1 - \sin(2t)\).

Step 1: Expand the Left Side

The left side of the given equation is \((\sin t - \cos t)^2\). To expand it, apply the square of a binomial formula: \((a-b)^2 = a^2 - 2ab + b^2\). This gives us: \[\sin^2 t - 2\sin t \cos t + \cos^2 t.\]

Step 2: Simplify Using a Trigonometric Identity

Recognize the identity \(\sin^2 t + \cos^2 t = 1\). Substitute it into the expression obtained from Step 1: \[1 - 2\sin t \cos t.\] This replaces \(\sin^2 t + \cos^2 t\) with 1.

Step 3: Express the Double Angle Formula

Recall the double angle formula for sine: \(\sin(2t) = 2\sin t \cos t\). Using this, we rewrite \(-2\sin t \cos t\) as \(-\sin(2t)\). So the expression becomes: \[1 - \sin(2t).\]

Step 4: Verify the Identity

Now both sides of the equation \((\sin t - \cos t)^2 = 1 - \sin(2t)\) match. Therefore, the given identity is proven to be true.

Key Concepts

Binomial Formula

The binomial formula is a fundamental concept in algebra that helps expand expressions raised to a power. For two terms, \(a\) and \(b\), the square of their binomial is denoted as \((a-b)^2\), which expands to \(a^2 - 2ab + b^2\). In our exercise, this formula was used to expand \((\sin t - \cos t)^2\). Applying the binomial formula allowed us to split the expression into individual terms: \(\sin^2 t - 2\sin t \cos t + \cos^2 t\).
This method of expansion simplifies complex expressions, making it easier to identify trigonometric identities. Remember the classic form of\

• \(a^2\)
• \(-2ab\)
• \(b^2\)

Using the binomial formula efficiently in algebra can greatly simplify solving equations, especially when working with trigonometric identities. This expansion is often the first critical step towards simplifying complex expressions.

Double Angle Formula

Trigonometric identities often rely on formulas like the double angle formula. For sine, the double angle formula is \(\sin(2t) = 2\sin t \cos t\). This formula allows you to express trigonometric functions at twice an angle in terms of products of functions at the original angle.
In the exercise, we saw that \(-2\sin t \cos t\) could be rewritten using the double angle formula as \(-\sin(2t)\). This transformation was crucial to simplifying the expression and proving that both sides of the identity are equal.

• It reduces complex trigonometric expressions into more manageable forms.
• It's particularly useful for solving trigonometric equations and proving identities.

Understanding the double angle formula can significantly ease calculations involving trigonometric identities. By recognizing patterns in the expression and appropriately applying the formula, complex identities can be proven easily.

Sine Function

The sine function, one of the primary trigonometric functions, relates an angle of a right triangle to the ratio of the length of the opposite side over the hypotenuse. It is a fundamental component of the unit circle and periodic phenomena.
Here are key properties of the sine function:

• The range of sine is from -1 to 1.
• Sine is periodic with a period of \(2\pi\).
• At \(t = 0\), \(\sin(t) = 0\).

In the original exercise, \(\sin(t)\) was part of the expression that was expanded and simplified to demonstrate the identity. Remember:

• The sine function is odd, meaning \(\sin(-t) = -\sin(t)\).
• It has symmetry about the origin.

Understanding sine is key to solving trigonometric identities as it often appears in various forms, such as in the double angle formula seen in this exercise.

Cosine Function

The cosine function is another primary trigonometric function and is very similar to the sine function. It relates an angle of a right triangle to the ratio of the length of the adjacent side over the hypotenuse. Like sine, cosine is part of the fundamental trigonometric functions within the unit circle.
Key properties include:

• The range is from -1 to 1, identical to that of sine.
• Cosine is an even function, meaning \(\cos(-t) = \cos(t)\).
• It is also periodic with a period of \(2\pi\).

In the exercise, \(\cos(t)\) was used together with \(\sin(t)\) to form expressions that were manipulated using the binomial and double angle formulas. One of the key identities used was \(\sin^2 t + \cos^2 t = 1\), which combines the two functions in an identity that helps simplify expressions.

• Cosine plays a crucial role in symmetry and transformations within trigonometric identities.
• Deep understanding of cosine's properties can immensely aid in solving complex trigonometric equations.