Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ (\sin t+\cos t)^{2}=1+\sin 2 t $$

Step-by-Step Solution

Answer

The identity is verified; both sides equal \(1 + \sin 2t\).

Step 1: Expand the Left Hand Side (LHS)

Let's first expand the square of the sum on the left side of the equation: \[(\sin t + \cos t)^2 = (\sin t + \cos t)(\sin t + \cos t) = \sin^2 t + 2\sin t\cos t + \cos^2 t.\]

Step 2: Apply the Pythagorean Identity

Recall the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\). Substitute this into our equation: \[\sin^2 t + \cos^2 t + 2\sin t\cos t = 1 + 2\sin t\cos t.\]

Step 3: Relate to the Right Hand Side (RHS)

The term \(2\sin t\cos t\) can be rewritten using the double angle identity for sine, \(\sin 2t = 2\sin t\cos t\). Therefore, we rewrite the equation as: \[1 + 2\sin t\cos t = 1 + \sin 2t.\]

Step 4: Verify Equality

Now both sides of the equation are identical: \[1 + \sin 2t = 1 + \sin 2t.\] Thus, the identity is verified as true.

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity that states:\[sin^2 t + cos^2 t = 1\]This identity is incredibly useful because it allows us to express one function in terms of the other. This means if you know the value of \(\sin t\), you can use this identity to find \(\cos t\), and vice versa. It's derived from the Pythagorean theorem applied to the unit circle, where the hypotenuse is 1.
Consider a right triangle inscribed in a unit circle:

• The length of the opposite side to the angle \(t\) is \(\sin t\).
• The length of the adjacent side to the angle \(t\) is \(\cos t\).
• The hypotenuse (radius of the circle) is 1.

This results in the equation \(\sin^2 t + \cos^2 t = 1\), aligning perfectly with the geometry of the triangle and circle.

Double Angle Identity

The double angle identity is another important trigonometric identity, which expresses trigonometric functions of double angles (like \(2t\)) in terms of single angles (such as \(t\)). Specifically, for the sine function, the identity is:\[sin 2t = 2 \sin t \cos t\]This identity is helpful in simplifying expressions involving the sine of a double angle. In this identity:

• \(\sin 2t\) is represented using \(\sin t\) and \(\cos t\).
• It highlights the relation between the sine of the double angle and the product of sine and cosine of the angle \(t\).

Double angle identities have numerous applications across different mathematical problems, including simplifying equations or solving trigonometric identities like the one given in this exercise.

Sine Function

The sine function, denoted as \(\sin\), is one of the most common trigonometric functions. It arises from the ratio of the length of the opposite side to the hypotenuse in a right triangle. In the context of the unit circle, the sine of an angle \(t\) is the y-coordinate of the point where the radius (or hypotenuse) at angle \(t\) intersects the circle.

• The sine function is periodic with a period of \(2\pi\), meaning \(\sin(t)=\sin(t+2\pi)\).
• Its range is the set of real numbers between -1 and 1.
• It's important for understanding oscillatory motion, waves, and alternating currents.

Connecting it to the exercise, the sine function helps express changes in the trigonometric setup as angles become larger, like using the double angle identity to transform \(2t\) terms into simpler components.