Problem 8 - Precalculus with Calculus Previews

Question

If \(\sin t=0.4,\) find all possible values of \(\cos t\)

Step-by-Step Solution

Answer

\(\cos t = 0.9165\) or \(\cos t = -0.9165\).

Step 1: Recall the Pythagorean identity

The most useful identity for relating sine and cosine is the Pythagorean identity: \[ \sin^2 t + \cos^2 t = 1 \]This identity helps us find the values of \(\cos t\) when \(\sin t\) is known.

Step 2: Substitute the given value into the identity

We are given \(\sin t = 0.4\). Substitute this value into the Pythagorean identity:\[ (0.4)^2 + \cos^2 t = 1 \]

Step 3: Simplify and solve for \(\\cos^2 t\)

Calculate \((0.4)^2 = 0.16\). Substitute it back into the equation:\[ 0.16 + \cos^2 t = 1 \]Subtract 0.16 from both sides to solve for \(\cos^2 t\):\[ \cos^2 t = 1 - 0.16 = 0.84 \]

Step 4: Solve for \(\\cos t\)

To find \(\cos t\), we need to take the square root of \(\cos^2 t\):\[ \cos t = \pm \sqrt{0.84} \]Thus, \(\cos t = \pm 0.9165\) (rounded to four decimal places).

Step 5: Interpret the results

Since the equation \(\sin t = 0.4\) does not specify any particular constraints on the quadrant, both positive and negative values for \(\cos t\) are valid.Hence, the possible values for \(\cos t\) are 0.9165 and -0.9165.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fascinating and fundamental concept in trigonometry. Derived from the Pythagorean Theorem, which relates the sides of a right triangle, this identity connects the sine and cosine functions in an elegant equation: \[ \sin^2 t + \cos^2 t = 1 \] This equation is valid for all angles \(t\), making it a powerful tool for solving problems where one of the trigonometric functions is known, and the other needs to be found.

• Why it's useful: With this identity, if you know either the sine or the cosine of an angle, you can easily determine the other. This is especially helpful in calculations involving angles in trigonometry.
• Key insight: This equation represents the unit circle — a circle with a radius of one centered at the origin on the coordinate plane — and shows that any point \((x, y)\) on the circle satisfies \(x^2 + y^2 = 1\).

Whenever faced with problems involving sine or cosine, the Pythagorean Identity is a good starting point.

Cosine Function

The cosine function, an essential part of trigonometry, describes the relationship between the angle and the ratio of the adjacent side over the hypotenuse in a right-angled triangle. Cosine is denoted as \(\cos\) and is prevalent in various applications in mathematics, physics, and engineering.The cosine function oscillates between -1 and 1 across its domain, repeating every \(2\pi\) radians.

• Positive and Negative Values: Cosine is positive in the first and fourth quadrants of the unit circle, and negative in the second and third. Thus, without additional constraints, both the positive and negative roots are valid solutions for \(\cos t\).
• Graph: The cosine function graph is a smooth wave, starting from a maximum of 1, decreasing through zero, to a minimum of -1, and back.

In our example, since \(\sin t = 0.4\), we calculate \(\cos t\) using the Pythagorean Identity and find \(\cos t = \pm 0.9165\). This reflects the dual nature of cosine's value based on the quadrant.

Sine Function

The sine function is another cornerstone of trigonometry, commonly represented as \(\sin\). It expresses the ratio of the opposite side over the hypotenuse in a right-angled triangle and is critical to understanding waves, oscillations, and circular motion.Just like the cosine, the sine function also oscillates between -1 and 1 over its period, although it starts from zero and then rises to its peak in the cycle.

• Behavior and Symmetry: Sine is positive in the first and second quadrants and negative in the third and fourth within a unit circle context. It has symmetry around the origin, known as an odd function.
• Visual Representation: The graph of the sine function is a smooth curve with a repeating pattern, much like a wave that starts from zero, ascends to 1, dipping back to zero, then continuing to -1.

When working with sine, as in our example with \(\sin t = 0.4\), the value guides the determination of the cosine through the Pythagorean Identity, linking the two functions in mathematical harmony.