Podcast
Questions and Answers
An angle in standard position has its vertex at the origin and its initial side along which axis?
An angle in standard position has its vertex at the origin and its initial side along which axis?
- Positive x-axis (correct)
- Negative x-axis
- Negative y-axis
- Positive y-axis
If the terminal side of an angle $\theta$ in standard position passes through the point $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ on the unit circle, what is the value of $\tan(\theta)$?
If the terminal side of an angle $\theta$ in standard position passes through the point $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ on the unit circle, what is the value of $\tan(\theta)$?
- 1 (correct)
- $\frac{\sqrt{2}}{2}$
- 0
- $\sqrt{2}$
Which of the following statements is true regarding the sine function in Quadrant II?
Which of the following statements is true regarding the sine function in Quadrant II?
- Sine is zero.
- Sine is negative.
- Sine is positive. (correct)
- Sine is undefined.
Given that $\sin(\theta) = \frac{1}{2}$ and $\theta$ is in Quadrant II, what is the value of $\cos(\theta)$?
Given that $\sin(\theta) = \frac{1}{2}$ and $\theta$ is in Quadrant II, what is the value of $\cos(\theta)$?
What is the reference angle for an angle of $240$ degrees?
What is the reference angle for an angle of $240$ degrees?
Which of the following trigonometric functions is an even function?
Which of the following trigonometric functions is an even function?
If $\cot(\theta) = -1$ and $\sin(\theta) > 0$, in which quadrant does $\theta$ lie?
If $\cot(\theta) = -1$ and $\sin(\theta) > 0$, in which quadrant does $\theta$ lie?
What is the value of $\sec(\frac{\pi}{3})$?
What is the value of $\sec(\frac{\pi}{3})$?
Which of the following identities is derived directly from the Pythagorean identity?
Which of the following identities is derived directly from the Pythagorean identity?
What is the period of the tangent function?
What is the period of the tangent function?
What are the coordinates of the point on the unit circle corresponding to an angle of $\frac{3\pi}{2}$ radians?
What are the coordinates of the point on the unit circle corresponding to an angle of $\frac{3\pi}{2}$ radians?
If $\sin(\theta) = 0$, which of the following could be a possible value for $\theta$ (in radians)?
If $\sin(\theta) = 0$, which of the following could be a possible value for $\theta$ (in radians)?
Given $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, at which point on the unit circle is $\tan(\theta)$ undefined?
Given $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, at which point on the unit circle is $\tan(\theta)$ undefined?
If the radius of a circle centered at the origin is 10, and a point on the circle corresponds to an angle $\theta$, what are the coordinates of that point in terms of $\theta$?
If the radius of a circle centered at the origin is 10, and a point on the circle corresponds to an angle $\theta$, what are the coordinates of that point in terms of $\theta$?
Which of the following describes the relationship between $\sin(\theta)$ and $\sin(-\theta)$?
Which of the following describes the relationship between $\sin(\theta)$ and $\sin(-\theta)$?
If $\cos(\theta) = -1$, what is the value of $\sin(\theta)$?
If $\cos(\theta) = -1$, what is the value of $\sin(\theta)$?
For what values of $\theta$ in the interval $[0, 2\pi)$ is $\csc(\theta)$ undefined?
For what values of $\theta$ in the interval $[0, 2\pi)$ is $\csc(\theta)$ undefined?
Given that $\sin(\theta) = \frac{\sqrt{2}}{2}$ and $\cos(\theta) = -\frac{\sqrt{2}}{2}$, find the value of $\theta$ in the interval $[0, 2\pi)$.
Given that $\sin(\theta) = \frac{\sqrt{2}}{2}$ and $\cos(\theta) = -\frac{\sqrt{2}}{2}$, find the value of $\theta$ in the interval $[0, 2\pi)$.
If an angle $\theta$ is in standard position and its terminal side passes through the point $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$ on the unit circle, what is the value of $\theta$ in degrees?
If an angle $\theta$ is in standard position and its terminal side passes through the point $(\frac{1}{2}, -\frac{\sqrt{3}}{2})$ on the unit circle, what is the value of $\theta$ in degrees?
Flashcards
Trigonometry
Trigonometry
Study of relationships between triangle sides/angles.
Unit Circle
Unit Circle
A circle with a radius of 1, centered at (0, 0).
Radians
Radians
A way to measure angles; a full rotation is 2Ï€.
Standard Position (Angle)
Standard Position (Angle)
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cos(θ)
cos(θ)
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sin(θ)
sin(θ)
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tan(θ)
tan(θ)
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csc(θ)
csc(θ)
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sec(θ)
sec(θ)
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cot(θ)
cot(θ)
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0 degrees (0 radians)
0 degrees (0 radians)
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30 degrees (Ï€/6 radians)
30 degrees (Ï€/6 radians)
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45 degrees (Ï€/4 radians)
45 degrees (Ï€/4 radians)
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60 degrees (Ï€/3 radians)
60 degrees (Ï€/3 radians)
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90 degrees (Ï€/2 radians)
90 degrees (Ï€/2 radians)
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Quadrantal Angles
Quadrantal Angles
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Pythagorean Identity
Pythagorean Identity
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Cosine (Even Function)
Cosine (Even Function)
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Sine (Odd Function)
Sine (Odd Function)
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Periodicity of Sine & Cosine
Periodicity of Sine & Cosine
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Study Notes
- Trigonometry is the study of the relationships between the sides and angles of triangles
- It's a fundamental branch of mathematics with applications in various fields like physics, engineering, and navigation
- The unit circle provides a visual and intuitive way to understand trigonometric functions
- It is a circle with a radius of one unit centered at the origin (0, 0) in the Cartesian coordinate system
Key Concepts
- Angle Measurement: Angles are measured in degrees or radians
- A full rotation around the circle is 360 degrees or 2Ï€ radians
- Important conversions: 180 degrees = π radians; 1 degree = π/180 radians; 1 radian = 180/π degrees
- Standard Position: An angle is in standard position when its vertex is at the origin and its initial side is along the positive x-axis
- Terminal Side: The side where the angle terminates after rotation
Coordinates on the Unit Circle
- For any point (x, y) on the unit circle corresponding to an angle θ:
- x = cos(θ)
- y = sin(θ)
- These relationships define the cosine and sine functions for all real numbers θ
- The coordinates (x, y) represent the cosine and sine of the angle θ, respectively
Trigonometric Functions
- Sine (sin θ): The y-coordinate of the point on the unit circle
- Cosine (cos θ): The x-coordinate of the point on the unit circle
- Tangent (tan θ): Defined as sin(θ) / cos(θ), which is y/x
- It represents the slope of the line from the origin to the point (x, y) on the unit circle
- Cosecant (csc θ): The reciprocal of sine, csc(θ) = 1 / sin(θ)
- Secant (sec θ): The reciprocal of cosine, sec(θ) = 1 / cos(θ)
- Cotangent (cot θ): The reciprocal of tangent, cot(θ) = cos(θ) / sin(θ) = x/y
Key Angles and Values
- 0 degrees (0 radians): (1, 0)
- sin(0) = 0, cos(0) = 1, tan(0) = 0
- 30 degrees (π/6 radians): (√3/2, 1/2)
- sin(π/6) = 1/2, cos(π/6) = √3/2, tan(π/6) = 1/√3 = √3/3
- 45 degrees (π/4 radians): (√2/2, √2/2)
- sin(π/4) = √2/2, cos(π/4) = √2/2, tan(π/4) = 1
- 60 degrees (π/3 radians): (1/2, √3/2)
- sin(π/3) = √3/2, cos(π/3) = 1/2, tan(π/3) = √3
- 90 degrees (Ï€/2 radians): (0, 1)
- sin(Ï€/2) = 1, cos(Ï€/2) = 0, tan(Ï€/2) = undefined
Quadrantal Angles
- Quadrantal angles are angles that lie on the axes (0, 90, 180, 270, 360 degrees)
- 0 degrees (0 radians): Point (1, 0)
- 90 degrees (Ï€/2 radians): Point (0, 1)
- 180 degrees (Ï€ radians): Point (-1, 0)
- 270 degrees (3Ï€/2 radians): Point (0, -1)
- 360 degrees (2Ï€ radians): Point (1, 0)
Signs of Trigonometric Functions by Quadrant
- Quadrant I (0 < θ < 90°): All trigonometric functions are positive
- Quadrant II (90° < θ < 180°): Sine (sin θ) is positive; cosine and tangent are negative
- Quadrant III (180° < θ < 270°): Tangent (tan θ) is positive; sine and cosine are negative
- Quadrant IV (270° < θ < 360°): Cosine (cos θ) is positive; sine and tangent are negative
Trigonometric Identities
- Pythagorean Identity: sin²(θ) + cos²(θ) = 1
- Derived Identities:
- 1 + tan²(θ) = sec²(θ)
- 1 + cot²(θ) = csc²(θ)
- Reciprocal Identities:
- csc(θ) = 1/sin(θ)
- sec(θ) = 1/cos(θ)
- cot(θ) = 1/tan(θ)
- Quotient Identities:
- tan(θ) = sin(θ)/cos(θ)
- cot(θ) = cos(θ)/sin(θ)
Even and Odd Functions
- Cosine is an even function: cos(-θ) = cos(θ)
- Sine is an odd function: sin(-θ) = -sin(θ)
- Tangent is an odd function: tan(-θ) = -tan(θ)
Periodic Properties
- Sine and cosine are periodic with a period of 2Ï€:
- sin(θ + 2π) = sin(θ)
- cos(θ + 2π) = cos(θ)
- Tangent is periodic with a period of π: tan(θ + π) = tan(θ)
Applications
- Solving Triangles: Finding unknown angles and sides of triangles
- Navigation: Determining direction and distance
- Physics: Analyzing oscillatory motion, waves
- Engineering: Designing structures, circuits
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