Problem 181 - Precalculus

Question

For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?

Step-by-Step Solution

Answer

Use \( \cos(\theta) = -\sqrt{1 - (\sin(\theta))^2} \) in quadrant II where cosine is negative.

Step 1: Understanding the Problem

We know an angle is in quadrant II. In this quadrant, the sine of the angle is positive, and the cosine of the angle is negative. We aim to determine the cosine given the sine of the angle.

Step 2: Recall a Trigonometric Identity

The identity we’ll use is Pythagoras' identity for sine and cosine: \[sin^2(\theta) + cos^2(\theta) = 1\]This identity relates the sine and cosine of any angle \(\theta\).

Step 3: Setup the Equation

Let the sine of the angle be \(\sin(\theta) = a\). Substitute this into the identity:\[a^2 + \cos^2(\theta) = 1\]

Step 4: Solve for Cosine Squared

Rearrange the equation to solve for \(\cos^2(\theta)\):\[\cos^2(\theta) = 1 - a^2\]

Step 5: Determine Cosine

To find \(\cos(\theta)\), take the square root of \(\cos^2(\theta)\):\[\cos(\theta) = \pm\sqrt{1 - a^2}\]

Step 6: Identify the Sign of Cosine

Since the angle is in quadrant II, the cosine is negative, so:\[\cos(\theta) = -\sqrt{1 - a^2}\]

Key Concepts

Quadrant II angles

In trigonometry, the Cartesian plane is divided into four quadrants. Quadrant II is where angles range from 90° to 180° or \(rac{\pi}{2}\) to \(rac{3\pi}{2}\) in radians. Knowing which quadrant an angle is in is essential for determining the signs of its trigonometric functions.

In Quadrant II:

• The sine function is positive.
• The cosine function is negative.
• The tangent function is also negative because it is the ratio of sine and cosine.

When solving problems, always consider which quadrant the angle belongs to. This will guide you in selecting the correct sign for trigonometric values. Quadrant II plays a crucial role in determining these signs and ensures that calculations are correct based on position.

Sine function

The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. In the unit circle, sine represents the y-coordinate of a point on the circle.
Key properties of the sine function include:

• It has a range from -1 to 1.
• It is an odd function, implying \(sin(-\theta)=-sin(\theta)\).
• Sine is positive in the first and second quadrants.

Understanding the sine function is foundational because it frequently appears in equations needing evaluation, especially when using trigonometric identities. Knowing that sine is positive in Quadrant II is important when applying this function to problems.

Cosine function

The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. In terms of the unit circle, cosine represents the x-coordinate.
Cosine characteristics to remember are:

• Cosine's range is from -1 to 1.
• It is an even function, meaning \(cos(-\theta)=cos(\theta)\).
• Cosine is positive in the first and fourth quadrants.

In Quadrant II, cosine is negative. This is pivotal when solving equations involving squares, as the resulting cosine value must be adjusted to fit the quadrant's sign rules. Always apply this when determining cosine from the Pythagorean identity.

Pythagorean identity

The Pythagorean identity is a fundamental relation in trigonometry, particularly useful for relating sine and cosine. It states: \[sin^2(\theta) + cos^2(\theta) = 1\]
This identity allows you to determine the cosine of an angle if you know its sine, and vice versa. This is because if you have one value, you can rearrange the formula to solve for the other:

• To find \(cos^2(\theta)\), use \[cos^2(\theta) = 1 - sin^2(\theta)\]
• To find \(sin^2(\theta)\), use \[sin^2(\theta) = 1 - cos^2(\theta)\]

When working with angles in any quadrant, adjusting the sign of sine or cosine depending on their quadrant is crucial. For Quadrant II, where cosine is negative, the negative square root is chosen after applying the identity. This ensures all trigonometric values fit their positional contexts.