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Questions and Answers
If $\sin(\theta) = \frac{5}{13}$ and $\theta$ is in Quadrant II, what is the value of $\cos(\theta)$?
If $\sin(\theta) = \frac{5}{13}$ and $\theta$ is in Quadrant II, what is the value of $\cos(\theta)$?
- $-\frac{12}{13}$ (correct)
- $\frac{8}{13}$
- $-\frac{8}{13}$
- $\frac{12}{13}$
Which of the following is equivalent to the expression $\frac{\sin(2x)}{\sin(x)}$?
Which of the following is equivalent to the expression $\frac{\sin(2x)}{\sin(x)}$?
- $2\sin(x)$
- $\sin(x)$
- $2\cos(x)$ (correct)
- $\cos(x)$
What is the general solution for the equation $2\cos(x) - 1 = 0$?
What is the general solution for the equation $2\cos(x) - 1 = 0$?
- $x = \frac{\pi}{4} + 2\pi k$ and $x = \frac{7\pi}{4} + 2\pi k$, where k is an integer
- $x = \frac{\pi}{3} + 2\pi k$ and $x = \frac{5\pi}{3} + 2\pi k$, where k is an integer (correct)
- $x = \frac{\pi}{6} + 2\pi k$ and $x = \frac{11\pi}{6} + 2\pi k$, where k is an integer
- $x = \frac{\pi}{2} + 2\pi k$ and $x = \frac{3\pi}{2} + 2\pi k$, where k is an integer
A projectile is launched with an initial velocity $v_0$ at an angle $\theta$ with respect to the horizontal. Which expression represents the range of the projectile, assuming level ground and neglecting air resistance?
A projectile is launched with an initial velocity $v_0$ at an angle $\theta$ with respect to the horizontal. Which expression represents the range of the projectile, assuming level ground and neglecting air resistance?
What is the domain and range of the inverse cosine function, $\arccos(x)$?
What is the domain and range of the inverse cosine function, $\arccos(x)$?
Given that $\tan(x) = \frac{3}{4}$ and $x$ is in the first quadrant, find the value of $\sin(2x)$.
Given that $\tan(x) = \frac{3}{4}$ and $x$ is in the first quadrant, find the value of $\sin(2x)$.
Simplify the expression: $\frac{\sin^4(x) - \cos^4(x)}{\sin^2(x) - \cos^2(x)}$
Simplify the expression: $\frac{\sin^4(x) - \cos^4(x)}{\sin^2(x) - \cos^2(x)}$
A wave is described by the equation $y = A\sin(Bx + C)$, where $A$, $B$, and $C$ are constants. How does the constant $C$ affect the wave?
A wave is described by the equation $y = A\sin(Bx + C)$, where $A$, $B$, and $C$ are constants. How does the constant $C$ affect the wave?
Which of the following expressions is equivalent to $\tan(\arccos(x))$?
Which of the following expressions is equivalent to $\tan(\arccos(x))$?
What is the value of $\arcsin(\sin(\frac{5\pi}{4}))$?
What is the value of $\arcsin(\sin(\frac{5\pi}{4}))$?
Flashcards
Sine (sin θ)
Sine (sin θ)
The ratio of the opposite side to the hypotenuse in a right triangle.
Cosine (cos θ)
Cosine (cos θ)
The ratio of the adjacent side to the hypotenuse in a right triangle.
Tangent (tan θ)
Tangent (tan θ)
The ratio of the opposite side to the adjacent side in a right triangle; also sin θ / cos θ.
Trigonometric identities
Trigonometric identities
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Pythagorean Identity (1)
Pythagorean Identity (1)
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Sum Formula for Sine
Sum Formula for Sine
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Unit Circle
Unit Circle
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Trigonometry in Projectile Motion
Trigonometry in Projectile Motion
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Inverse Sine (arcsin or sin⁻¹)
Inverse Sine (arcsin or sin⁻¹)
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Inverse Cosine (arccos or cos⁻¹)
Inverse Cosine (arccos or cos⁻¹)
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Study Notes
- Trigonometry studies the correlation between triangle side lengths and angles
- It's crucial for fields like physics, engineering, and navigation
- Trigonometric functions link triangle angles to ratios of its sides
Trigonometric Functions
- Sine (sin θ) in a right triangle equals the opposite side divided by the hypotenuse
- Cosine (cos θ) equals the adjacent side divided by the hypotenuse
- Tangent (tan θ) equals the opposite side divided by the adjacent side, and also equals sin θ / cos θ
- Cosecant (csc θ) is the reciprocal of sine: csc θ = 1 / sin θ
- Secant (sec θ) is the reciprocal of cosine: sec θ = 1 / cos θ
- Cotangent (cot θ) is the reciprocal of tangent: cot θ = 1 / tan θ = cos θ / sin θ
- These functions exhibit periodicity, with values repeating after consistent intervals
Identities
- Trigonometric identities are equations involving trigonometric functions that hold true for all variable values
- Pythagorean Identities:
- sin² θ + cos² θ = 1
- 1 + tan² θ = sec² θ
- 1 + cot² θ = csc² θ
- Sum and Difference Formulas:
- sin(A + B) = sin A cos B + cos A sin B
- sin(A - B) = sin A cos B - cos A sin B
- cos(A + B) = cos A cos B - sin A sin B
- cos(A - B) = cos A cos B + sin A sin B
- tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
- tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
- Double Angle Formulas:
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos² θ - sin² θ = 2 cos² θ - 1 = 1 - 2 sin² θ
- tan 2θ = (2 tan θ) / (1 - tan² θ)
- Half Angle Formulas:
- sin (θ/2) = ±√((1 - cos θ) / 2)
- cos (θ/2) = ±√((1 + cos θ) / 2)
- tan (θ/2) = ±√((1 - cos θ) / (1 + cos θ)) = (sin θ) / (1 + cos θ) = (1 - cos θ) / (sin θ)
Trigonometric Equations
- Trigonometric equations feature trigonometric functions, solved by finding variable values that satisfy the equation
- Steps to solve trigonometric equations:
- Use algebraic methods to isolate the trigonometric function
- Identify angles satisfying the equation within a specific interval
- Account for the functions' periodicity to find all possible solutions
- For example, sin x = 0.5 has solutions x = π/6 + 2πk and x = 5π/6 + 2πk, where k is an integer
Unit Circle
- The unit circle has a radius of 1, centered at the origin (0,0)
- It visually represents and helps in understanding trigonometric functions for all real number inputs
- For angle θ, coordinates where the angle's terminal side intersects the circle are (cos θ, sin θ)
- The unit circle aids in determining trigonometric function values for common angles like 0, π/6, π/4, π/3, π/2, π, etc.
- It shows how trigonometric functions' signs vary by quadrant:
- Quadrant I (0 to π/2): Functions are positive
- Quadrant II (π/2 to π): Sine is positive; others are negative
- Quadrant III (π to 3π/2): Tangent is positive; others are negative
- Quadrant IV (3π/2 to 2π): Cosine is positive; others are negative
Applications in Physics
- Projectile Motion: Trigonometry helps analyze projectile motion, like range, height, and flight time
- Simple Harmonic Motion: Trigonometric functions describe oscillations in pendulums or mass-spring systems
- Waves: Trigonometric functions model wave phenomena, including sound, light, and water waves
- Optics: It is used to analyze light behavior, covering reflection, refraction, and diffraction
- Statics and Dynamics: Forces and components are analyzed with trigonometric functions
- Electromagnetism: Trigonometric functions are employed to describe alternating currents and electromagnetic waves
Inverse Trigonometric Functions
- These functions invert trigonometric functions, finding angles corresponding to trigonometric ratios
- Inverse Sine (arcsin or sin⁻¹): Delivers the angle with a given sine value
- Domain: [-1, 1], Range: [-π/2, π/2]
- Inverse Cosine (arccos or cos⁻¹): Delivers the angle with a given cosine value
- Domain: [-1, 1], Range: [0, π]
- Inverse Tangent (arctan or tan⁻¹): Delivers the angle with a given tangent value
- Domain: (-∞, ∞), Range: (-π/2, π/2)
- Inverse Cosecant (arccsc or csc⁻¹): Delivers the angle with a given cosecant value
- Domain: (-∞, -1] ∪ [1, ∞], Range: [-π/2, 0) ∪ (0, π/2]
- Inverse Secant (arcsec or sec⁻¹): Delivers the angle with a given secant value
- Domain: (-∞, -1] ∪ [1, ∞], Range: [0, π/2) ∪ (π/2, π]
- Inverse Cotangent (arccot or cot⁻¹): Delivers the angle with a given cotangent value
- Domain: (-∞, ∞), Range: (0, π)
- Consider the range when using inverse trigonometric functions for accurate angle determination
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