Problem 58 - Contemporary Precalculus

Question

In Exercises \(57-62,\) assume that the terminal side of an angle of t radians in standard position lies in the given quadrant on the given straight line. Find sin \(t,\) cos \(t,\) tan \(t .\) [Hint: Find \(a\) point on the terminal side of the angle.\(]\) Quadrant III; line with equation \(2 y-5 x=0\)

Step-by-Step Solution

Answer

Answer: sin t = \(-\frac{5}{\sqrt{29}}\), cos t = \(-\frac{2}{\sqrt{29}}\), tan t = \(\frac{5}{2}\)

Step 1: Determine a point on the terminal side of the angle

Let's find any point that lies on both graph Quadrant III and the given line \(2y - 5x = 0\). Since we are working in Quadrant III, we know both x and y values will be negative. We can choose a suitable x value and solve for y. Let's take x = -1: \(2y - 5(-1) = 0\) \(2y + 5 = 0\) \(2y = -5\) \(y = -\frac{5}{2}\) So, the point on the terminal side of the angle is \((-1, -\frac{5}{2})\).

Step 2: Calculate the length of the hypotenuse

We can determine the length of the legs of our triangle by using the absolute values of the coordinates of our point. The legs of the triangle have lengths \(|x| = 1\) and \(|y| = \frac{5}{2}\). The hypotenuse, r, can be found using the Pythagorean theorem: \(r^2 = x^2 + y^2\) \(r^2 = (-1)^2 + \left(-\frac{5}{2}\right)^2\) \(r^2 = 1 + \frac{25}{4}\) \(r^2 = \frac{29}{4}\) \(r = \frac{\sqrt{29}}{2}\)

Step 3: Find sin t, cos t, and tan t using the triangle properties

Now, we can find sin t, cos t, and tan t using the ratios given by the lengths of the sides we have determined: sin t = \(\frac{y}{r} = \frac{-\frac{5}{2}}{\frac{\sqrt{29}}{2}} = -\frac{5}{\sqrt{29}}\) cos t = \(\frac{x}{r} = \frac{-1}{\frac{\sqrt{29}}{2}} = -\frac{2}{\sqrt{29}}\) tan t = \(\frac{\text{sin t}}{\text{cos t}} = \frac{-\frac{5}{\sqrt{29}}}{-\frac{2}{\sqrt{29}}} = \frac{5}{2}\) Hence, the sin, cos, and tan of the angle t respectively are: sin t = \(-\frac{5}{\sqrt{29}}\) cos t = \(-\frac{2}{\sqrt{29}}\) tan t = \(\frac{5}{2}\)

Key Concepts

Trigonometry in standard position

Understanding trigonometry in standard position is essential for solving trigonometric problems effectively. When we say an angle is in 'standard position,' we mean its vertex is at the origin of a coordinate system, and its initial side lies on the positive x-axis. The angle's terminal side is determined by the amount of rotation from the initial side.

For example, if we rotate an initial line counterclockwise to rest in the third quadrant, the terminal side of the angle would lie in that quadrant. The standard position is important because it allows us to use the coordinate plane to define the trigonometric functions sine, cosine, and tangent as ratios involving the lengths of sides of a right triangle formed by dropping a perpendicular from the point on the terminal side to the x-axis.

Solving trigonometric ratios

Solving trigonometric ratios involves finding the values of sine (sin), cosine (cos), and tangent (tan) for a given angle. These ratios are based on the relationships between the sides of a right-angled triangle. Specifically:

• Sin is the ratio of the opposite side to the hypotenuse.
• Cos is the ratio of the adjacent side to the hypotenuse.
• Tan is the ratio of the opposite side to the adjacent side.

To calculate these ratios, one must have a clear understanding of the angle's position in the coordinate system and the signs of the x and y coordinates—which change depending on the quadrant. For angles in standard position, this means identifying a point on the terminal side of the angle and using its coordinates to compute the ratios, considering the quadrant's specific sign conventions.

Pythagorean theorem application

The Pythagorean theorem is a foundational principle in trigonometry used to determine the relationships between the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Expressed as an equation, it is:
\[ a^2 + b^2 = c^2 \]
where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the length of the hypotenuse. This theorem enables us to calculate the hypotenuse's length when its legs' lengths are known—a critical step when finding trigonometric ratios. In our solution example, the coordinates given by the point \((-1, -\frac{5}{2})\) determine the lengths of the legs, and consequently, we apply the Pythagorean theorem to find the hypotenuse, a necessary component for computing sine, cosine, and tangent values.

Quadrant III trigonometry

In trigonometry, the coordinate plane is divided into four quadrants, each with unique properties regarding the signs of trigonometric functions. For Quadrant III, where both x and y coordinates are negative, the implication for trigonometric functions is specific: both sine and cosine are negative due to the negative y (opposite) and x (adjacent) values, but tangent, being the ratio of sine to cosine (opposite to adjacent), is positive since the negative signs cancel out.

Remembering the phrase 'All Students Take Calculus' can be helpful. It stands for:

• All - All trigonometric functions are positive in Quadrant I
• Students - Only Sine is positive in Quadrant II
• Take - Only Tangent is positive in Quadrant III
• Calculus - Only Cosine is positive in Quadrant IV

This mnemonic helps remember which functions are positive in each quadrant. In our exercise, knowing that the angle lies in Quadrant III, we can predict that sin and cos will yield negative results, and tan will be positive before we even calculate the values using the trigonometric ratios.