Trigonometry Beyond Right Triangles

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Questions and Answers

Explain how the unit circle simplifies understanding trigonometric functions for angles beyond those found in right triangles (i.e., angles greater than 90 degrees).

The unit circle extends the definitions of trigonometric functions to all real numbers by relating angles to coordinates on the circle, where cosine and sine are the x and y coordinates, respectively. It visualizes the sign and value changes of the functions as the angle rotates around the circle.

If $\sin(\theta) = -\frac{1}{2}$ and $\theta$ is in the third quadrant, what is the value of $\cos(\theta)$?

$\cos(\theta) = -\frac{\sqrt{3}}{2}$

Describe how the tangent function is represented on the unit circle and how its value changes as the angle approaches 90 degrees.

Tangent is represented as the ratio of the y-coordinate to the x-coordinate ($\frac{\sin(\theta)}{\cos(\theta)}$). As the angle approaches 90 degrees, the x-coordinate approaches 0, causing the tangent value to approach infinity.

Explain the relationship between the angle of elevation and the angle of depression when observing an object from two different points (e.g., from a building to a boat and from the boat to the building).

The angle of elevation from one point to another is equal to the angle of depression from the second point to the first. They are alternate interior angles between parallel lines. More specifically, the 'horizontal line of sight' from each point. Signup and view all the answers

A ladder leans against a wall, forming an angle of elevation of 60 degrees with the ground. If the foot of the ladder is 4 meters away from the wall, how high up the wall does the ladder reach? Express your answer using trigonometric functions.

Height = $4 \cdot \tan(60^{\circ})$ or $4\sqrt{3}$ meters Signup and view all the answers

Explain why $\sin(\theta)$ and $\csc(\theta)$ always have the same sign, while $\tan(\theta)$ and $\cot(\theta)$ also always have the same sign.

$\csc(\theta)$ is the reciprocal of $\sin(\theta)$ ($\csc(\theta) = \frac{1}{\sin(\theta)}$), and $\cot(\theta)$ is the reciprocal of $\tan(\theta)$ ($\cot(\theta) = \frac{1}{\tan(\theta)}$). Reciprocals have the same sign as the original value. Signup and view all the answers

Describe how the signs of $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$ change across the four quadrants of the unit circle.

Quadrant I: All positive. Quadrant II: $\sin(\theta)$ positive, $\cos(\theta)$ and $\tan(\theta)$ negative. Quadrant III: $\tan(\theta)$ positive, $\sin(\theta)$ and $\cos(\theta)$ negative. Quadrant IV: $\cos(\theta)$ positive, $\sin(\theta)$ and $\tan(\theta)$ negative. Signup and view all the answers

A surveyor stands 100 meters from the base of a tall building. The angle of elevation to the top of the building is 30 degrees. What is the height of the building? Write your answer using trigonometric functions.

Height = $100 \cdot \tan(30^{\circ})$ or $\frac{100}{\sqrt{3}}$ meters. Simplifying, $\frac{100\sqrt{3}}{3}$ meters Signup and view all the answers

Explain how you would determine the reference angle for an angle of 240 degrees and why reference angles are useful in trigonometry.

The reference angle is $240^{\circ} - 180^{\circ} = 60^{\circ}$. Reference angles are useful because they allow you to find the trigonometric values of any angle by relating it to an angle in the first quadrant. Signup and view all the answers

If $\cos(\theta) = \frac{\sqrt{2}}{2}$ and $\theta$ is in the fourth quadrant, find the values of $\sin(\theta)$ and $\tan(\theta)$.

$\sin(\theta) = -\frac{\sqrt{2}}{2}$ and $\tan(\theta) = -1$ Signup and view all the answers

Describe the transformations of the sine function, $y = A\sin(Bx + C) + D$, and how each parameter affects the graph.

A: Amplitude (vertical stretch). B: Affects the period (horizontal compression/stretch). C: Phase shift (horizontal shift). D: Vertical shift. Signup and view all the answers

Explain how the Pythagorean identity, $\sin^2(\theta) + \cos^2(\theta) = 1$, relates to the unit circle.

On the unit circle, $\sin(\theta)$ and $\cos(\theta)$ represent the y and x coordinates, respectively, of a point on the circle. The Pythagorean identity is a direct application of the Pythagorean theorem ($a^2 + b^2 = c^2$) with $c = 1$ (the radius of the unit circle). Signup and view all the answers

From the top of a lighthouse 20 meters high, the angle of depression to a boat is 45 degrees. How far is the boat from the base of the lighthouse?

20 meters Signup and view all the answers

Given that $\tan(\theta) = \frac{3}{4}$ and $\theta$ is in the first quadrant, determine the values of $\sin(\theta)$ and $\cos(\theta)$.

$\sin(\theta) = \frac{3}{5}$ and $\cos(\theta) = \frac{4}{5}$ Signup and view all the answers

Explain the difference between an angle measured in degrees and an angle measured in radians, and provide the conversion formula between them.

Degrees are a unit of angular measure where a full circle is 360 degrees, while radians are based on the radius of the circle, where a full circle is $2\pi$ radians. Conversion: degrees = radians * $\frac{180}{\pi}$; radians = degrees * $\frac{\pi}{180}$ Signup and view all the answers

How can the unit circle be used to find the values of trigonometric functions for angles greater than $360^{\circ}$ or less than $0^{\circ}$?

For angles greater than $360^{\circ}$ or less than $0^{\circ}$, find the coterminal angle within the range of $0^{\circ}$ to $360^{\circ}$ by adding or subtracting multiples of $360^{\circ}$. Use the coordinates of the coterminal angle on the unit circle to determine the trigonometric function values. Signup and view all the answers

Describe a real-world scenario where angles of elevation and depression are used together to solve a problem. Explain how these angles are related in your scenario.

Scenario: Two people are on top of buildings looking at each other. The angle of elevation from the shorter building to the taller building is related to the angle of depression from the taller building to the shorter building; they are equal because they are alternate interior angles formed by a transversal (line of sight) intersecting two parallel lines (horizontal lines). Signup and view all the answers

Determine the values of all six trigonometric functions for the angle $\theta = \frac{3\pi}{2}$.

$\sin(\frac{3\pi}{2}) = -1$, $\cos(\frac{3\pi}{2}) = 0$, $\tan(\frac{3\pi}{2})$ is undefined, $\csc(\frac{3\pi}{2}) = -1$, $\sec(\frac{3\pi}{2})$ is undefined, $\cot(\frac{3\pi}{2}) = 0$ Signup and view all the answers

Explain how the concept of similar triangles is related to the definitions of trigonometric functions in right triangles.

Trigonometric functions are defined as ratios of sides in a right triangle. Similar triangles have the same angles, so the ratios of corresponding sides are equal. This means that for a given angle, the trigonometric functions will have the same value, regardless of the size of the right triangle. Signup and view all the answers

An airplane is flying at an altitude of 1000 meters. The angle of depression from the airplane to a control tower is 20 degrees. What is the horizontal distance from the airplane to the control tower? Express your answer using trigonometric functions.

Horizontal distance = $1000 / \tan(20^{\circ})$ meters Signup and view all the answers

Flashcards

Unit Circle

Circle with a radius of 1, centered at the origin (0,0) in the Cartesian coordinate system.

cos θ on Unit Circle

For angle θ, x-coordinate of the point where the terminal side intersects the unit circle.