Problem 42 - Contemporary Precalculus

Question

In Exercises \(37-42,\) find \(\sin t,\) cos \(t,\) tan \(t\) when the terminal side of an angle of t radians in standard position passes through the given point. $$(-2, \pi)$$

Step-by-Step Solution

Answer

Answer: sin(t) = 𝜋 / √(4 + 𝜋^2) cos(t) = -2 / √(4 + 𝜋^2) tan(t) = 𝜋 / -2

Step 1: Find the hypotenuse length

To find the hypotenuse length (r) of the right triangle, we will use the distance formula: r = √(x^2 + y^2). In this case, x = -2 and y = 𝜋: r = √((-2)^2 + (𝜋)^2) r = √(4 + 𝜋^2)

Step 2: Calculate sin(t)

To find the sin(t), we can use the formula sin(t) = y/r, where y is the y-coordinate of the given point and r is the hypotenuse length: sin(t) = 𝜋 / √(4 + 𝜋^2)

Step 3: Calculate cos(t)

To find the cos(t), we can use the formula cos(t) = x/r, where x is the x-coordinate of the given point and r is the hypotenuse length: cos(t) = -2 / √(4 + 𝜋^2)

Step 4: Calculate tan(t)

To find the tan(t), we can use the formula tan(t) = sin(t) / cos(t): tan(t) = [𝜋 / √(4 + 𝜋^2)] / [-2 / √(4 + 𝜋^2)] To simplify this expression, we can multiply the numerator and denominator by √(4 + 𝜋^2): tan(t) = [(𝜋 / √(4 + 𝜋^2)) * √(4 + 𝜋^2)] / [(-2 / √(4 + 𝜋^2)) * √(4 + 𝜋^2)] tan(t) = (𝜋 * √(4 + 𝜋^2)) / (-2 * √(4 + 𝜋^2)) Now, we can cancel out the common term √(4 + 𝜋^2) from the numerator and the denominator: tan(t) = 𝜋 / -2 So, we have found that: sin(t) = 𝜋 / √(4 + 𝜋^2) cos(t) = -2 / √(4 + 𝜋^2) tan(t) = 𝜋 / -2

Key Concepts

Trigonometric Functions

Understanding trigonometric functions is essential when analyzing angles and their relationships with right triangles. These functions include sine (sin), cosine (cos), and tangent (tan), which are defined using the ratios of the sides of a right triangle or, in a broader context, the coordinates of a point on the unit circle.

For example, the sine of an angle is the ratio of the opposite side to the hypotenuse, while cosine represents the adjacent side over the hypotenuse, and tangent is the opposite over the adjacent. In the given exercise, these definitions translate into the formulas:

• For sine: \(\sin(t) = \frac{y}{r}\)
• For cosine: \(\cos(t) = \frac{x}{r}\)
• For tangent: \(\tan(t) = \frac{\sin(t)}{\cos(t)}\)

Using coordinates \((-2, \pi)\) and the calculated hypotenuse, we apply these formulas to find the trigonometric ratios of angle t. The precision in understanding and applying these formulas is key in precalculus trigonometry.

Terminal Side of an Angle

In trigonometry, every angle in standard position has two sides: the initial side and the terminal side. The initial side is always on the positive x-axis, while the terminal side is where the angle 'ends' after rotation. To work with angles in trigonometry, we often draw them in a coordinate system where their terminal side intersects a circle centered at the origin.

This intersection point provides crucial information for calculating trigonometric functions, as seen in our exercise. By knowing that the terminal side of angle t passes through point \((-2, \pi)\), we're equipped to use the coordinates of this point as the x and y values necessary to find the sine, cosine, and tangent of the angle. The exact position of the terminal side is vital, as it affects the sign and value of trigonometric functions.

Standard Position

Angles in standard position are a fundamental concept in trigonometry. An angle is said to be in standard position when its vertex is at the origin of the coordinate system, and the initial side coincides with the positive x-axis. From this fixed starting point, the angle opens counter-clockwise for positive angles and clockwise for negative angles, with the terminal side determining the angle's measure.

In our textbook problem, we are told that the angle t is in standard position, and hence we use the coordinates of the point where the terminal side intersects the curve to find the desired trigonometric functions. Remember, the standard position allows us to apply consistent rules and formulas to compute trigonometric values for angles, ultimately leading to a deeper understanding of their properties.