Problem 73
Question
Find the exact values of the six trigonometric functions of \(\theta\) if \(\theta\) is in standard position and \(P\) is on the terminal side. $$P(-2,-5)$$
Step-by-Step Solution
Verified Answer
The six trigonometric functions are: \(\sin \theta = \frac{-5}{\sqrt{29}}\), \(\cos \theta = \frac{-2}{\sqrt{29}}\), \(\tan \theta = \frac{5}{2}\), \(\csc \theta = \frac{\sqrt{29}}{-5}\), \(\sec \theta = \frac{\sqrt{29}}{-2}\), \(\cot \theta = \frac{2}{5}\).
1Step 1: Determine the Radius
First, calculate the radius (or hypotenuse) using the Pythagorean theorem. This will be equivalent to the distance from the origin (0,0) to the point \(P(-2,-5)\). Apply the formula: \[r = \sqrt{x^2 + y^2}\]Substitute \(x = -2\) and \(y = -5\):\[r = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}\]
2Step 2: Calculate Sine and Cosecant
The sine of an angle \(\theta\) is defined as \(\sin \theta = \frac{y}{r}\), and the cosecant is its reciprocal \(\csc \theta = \frac{r}{y}\).Substitute the values:\[\sin \theta = \frac{-5}{\sqrt{29}}, \quad \csc \theta = \frac{\sqrt{29}}{-5}\]
3Step 3: Calculate Cosine and Secant
The cosine of an angle is \(\cos \theta = \frac{x}{r}\), and the secant is \(\sec \theta = \frac{r}{x}\).Substitute the values:\[\cos \theta = \frac{-2}{\sqrt{29}}, \quad \sec \theta = \frac{\sqrt{29}}{-2}\]
4Step 4: Calculate Tangent and Cotangent
The tangent of an angle \(\theta\) is \(\tan \theta = \frac{y}{x}\), and the cotangent is \(\cot \theta = \frac{x}{y}\).Substitute the values:\[\tan \theta = \frac{-5}{-2} = \frac{5}{2}, \quad \cot \theta = \frac{-2}{-5} = \frac{2}{5}\]
Key Concepts
Right TriangleSine and CosinePythagorean TheoremAngles in Standard Position
Right Triangle
A right triangle is a triangle that has one angle exactly equal to 90 degrees. In such triangles, the two sides that form the right angle are called the 'legs', and the side opposite the right angle is known as the 'hypotenuse'. Understanding right triangles is crucial when dealing with trigonometric functions, as many of these functions are based on the relationships between the angles and sides of these triangles.
When working with trigonometric problems, you often use the framework of the right triangle to relate angles to side lengths. For a triangle with a point like \(P(-2,-5)\), you can imagine the vertical and horizontal lines dropping down and across to the point forming the triangle in the coordinate plane. This helps in visualizing the triangle's orientation and calculating its dimensions.
When working with trigonometric problems, you often use the framework of the right triangle to relate angles to side lengths. For a triangle with a point like \(P(-2,-5)\), you can imagine the vertical and horizontal lines dropping down and across to the point forming the triangle in the coordinate plane. This helps in visualizing the triangle's orientation and calculating its dimensions.
Sine and Cosine
Sine and cosine are two fundamental trigonometric functions that relate the angles of a right triangle to the lengths of its sides.
In the coordinate system, when a point like \(P(-2,-5)\) lies on the terminal side of an angle in standard position:
In the coordinate system, when a point like \(P(-2,-5)\) lies on the terminal side of an angle in standard position:
- The sine function is defined as the ratio of the length of the opposite side to the hypotenuse. For point \(P(-2, -5)\), \( \sin \theta = \frac{y}{r}= \frac{-5}{\sqrt{29}}\).
- The cosine function is defined as the ratio of the length of the adjacent side to the hypotenuse. Here, \(\cos \theta = \frac{x}{r} = \frac{-2}{\sqrt{29}}\).
Pythagorean Theorem
The Pythagorean theorem is a key concept in trigonometry, especially with right triangles. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
\[c^2 = a^2 + b^2\]
When using the theorem with a point like \(P(-2,-5)\), the radius \(r\) can be determined as the hypotenuse of the triangle formed with the x and y-axis. Here, applying the Pythagorean theorem gives:
\[r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}\]
This calculation of the hypotenuse is crucial for determining exact values of trigonometric functions.
\[c^2 = a^2 + b^2\]
When using the theorem with a point like \(P(-2,-5)\), the radius \(r\) can be determined as the hypotenuse of the triangle formed with the x and y-axis. Here, applying the Pythagorean theorem gives:
\[r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}\]
This calculation of the hypotenuse is crucial for determining exact values of trigonometric functions.
Angles in Standard Position
An angle is said to be in standard position when its vertex is at the origin of a coordinate plane, and its initial side lies along the positive x-axis. The angle sweeps counterclockwise to intersect with a point. For instance, if point \(P(-2,-5)\) lies on the terminal side, the angle \(\theta\) is in the fourth quadrant.
Understanding angles in standard position is important as it helps in determining the exact values of trigonometric functions based on the quadrant they fall in. Each quadrant affects the sign of the trigonometric functions:
Understanding angles in standard position is important as it helps in determining the exact values of trigonometric functions based on the quadrant they fall in. Each quadrant affects the sign of the trigonometric functions:
- First Quadrant: all functions are positive.
- Second Quadrant: sine is positive.
- Third Quadrant: tangent is positive.
- Fourth Quadrant: cosine is positive.
Other exercises in this chapter
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