Trigonometry - ACT Math
Card 1 of 30
What is the sine of a $30^\text{o}$ angle?
What is the sine of a $30^\text{o}$ angle?
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$\frac{1}{2}$. This is a special angle value from the 30-60-90 triangle.
$\frac{1}{2}$. This is a special angle value from the 30-60-90 triangle.
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What is the cosine of $180^\text{o}$?
What is the cosine of $180^\text{o}$?
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-1. At 180°, we're on the negative x-axis where x = -1.
-1. At 180°, we're on the negative x-axis where x = -1.
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What is the cosecant of a 0-degree angle?
What is the cosecant of a 0-degree angle?
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Undefined. Division by zero makes cosecant undefined.
Undefined. Division by zero makes cosecant undefined.
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What is the tangent of a 90-degree angle?
What is the tangent of a 90-degree angle?
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Undefined. Division by zero makes tangent undefined.
Undefined. Division by zero makes tangent undefined.
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What is the tangent of $180^\text{o}$?
What is the tangent of $180^\text{o}$?
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- Since $\sin(180°) = 0$ and $\cos(180°) = -1$.
- Since $\sin(180°) = 0$ and $\cos(180°) = -1$.
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State the formula for converting degrees to radians.
State the formula for converting degrees to radians.
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$\theta \text{ (radians)} = \theta \text{ (degrees)} \times \frac{\pi}{180}$. Multiply by $\frac{\pi}{180}$ to convert degrees to radians.
$\theta \text{ (radians)} = \theta \text{ (degrees)} \times \frac{\pi}{180}$. Multiply by $\frac{\pi}{180}$ to convert degrees to radians.
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State the Pythagorean identity for tangent and secant.
State the Pythagorean identity for tangent and secant.
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$1 + \text{tan}^2 \theta = \text{sec}^2 \theta$. Derived from the Pythagorean identity and reciprocals.
$1 + \text{tan}^2 \theta = \text{sec}^2 \theta$. Derived from the Pythagorean identity and reciprocals.
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Convert $\frac{\text{π}}{2}$ radians to degrees.
Convert $\frac{\text{π}}{2}$ radians to degrees.
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90 degrees. $\frac{\pi}{2}$ radians equals 90° by definition.
90 degrees. $\frac{\pi}{2}$ radians equals 90° by definition.
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What is the cotangent of a 45-degree angle?
What is the cotangent of a 45-degree angle?
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- At 45°, sine equals cosine, so their ratio is 1.
- At 45°, sine equals cosine, so their ratio is 1.
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What is the cotangent of a 0-degree angle?
What is the cotangent of a 0-degree angle?
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Undefined. Division by zero makes cotangent undefined.
Undefined. Division by zero makes cotangent undefined.
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What is the secant of a 90-degree angle?
What is the secant of a 90-degree angle?
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Undefined. Division by zero makes secant undefined.
Undefined. Division by zero makes secant undefined.
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What is the cosecant of a 30-degree angle?
What is the cosecant of a 30-degree angle?
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$2$. Cosecant is $\frac{1}{\sin(30°)} = \frac{1}{1/2}$.
$2$. Cosecant is $\frac{1}{\sin(30°)} = \frac{1}{1/2}$.
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What is the value of $\text{tan}(0^\text{o})$?
What is the value of $\text{tan}(0^\text{o})$?
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- At 0°, opposite side has zero length.
- At 0°, opposite side has zero length.
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What are the right-triangle definitions of $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$?
What are the right-triangle definitions of $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$?
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$\sin=\frac{\text{opp}}{\text{hyp}},\ \cos=\frac{\text{adj}}{\text{hyp}},\ \tan=\frac{\text{opp}}{\text{adj}}$. SOH-CAH-TOA: ratios of opposite, adjacent sides to hypotenuse.
$\sin=\frac{\text{opp}}{\text{hyp}},\ \cos=\frac{\text{adj}}{\text{hyp}},\ \tan=\frac{\text{opp}}{\text{adj}}$. SOH-CAH-TOA: ratios of opposite, adjacent sides to hypotenuse.
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What is the formula for the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ in the plane?
What is the formula for the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ in the plane?
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$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Pythagorean theorem applied to coordinate differences.
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Pythagorean theorem applied to coordinate differences.
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What is the sine of an angle whose secant is 2?
What is the sine of an angle whose secant is 2?
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$\frac{\sqrt{3}}{2}$. If $\sec = 2$, then $\cos = \frac{1}{2}$, use Pythagorean identity.
$\frac{\sqrt{3}}{2}$. If $\sec = 2$, then $\cos = \frac{1}{2}$, use Pythagorean identity.
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What is the reciprocal of cosine?
What is the reciprocal of cosine?
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Secant. Reciprocal function: $\sec(\theta) = \frac{1}{\cos(\theta)}$.
Secant. Reciprocal function: $\sec(\theta) = \frac{1}{\cos(\theta)}$.
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What is the cosine of a $90^\text{o}$ angle?
What is the cosine of a $90^\text{o}$ angle?
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- At 90°, the adjacent side is zero.
- At 90°, the adjacent side is zero.
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State the Pythagorean identity involving sine and cosine.
State the Pythagorean identity involving sine and cosine.
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$\text{sin}^2 \theta + \text{cos}^2 \theta = 1$. This fundamental identity comes from the Pythagorean theorem.
$\text{sin}^2 \theta + \text{cos}^2 \theta = 1$. This fundamental identity comes from the Pythagorean theorem.
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Find the exact value of $\sin(\frac{\pi}{6})$.
Find the exact value of $\sin(\frac{\pi}{6})$.
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$\frac{1}{2}$. This is $\sin(30^\circ)$, a special angle value.
$\frac{1}{2}$. This is $\sin(30^\circ)$, a special angle value.
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What is the period of $y = \cos(2x)$?
What is the period of $y = \cos(2x)$?
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$\pi$. Period formula: $\frac{2\pi}{|b|}$ where $b = 2$.
$\pi$. Period formula: $\frac{2\pi}{|b|}$ where $b = 2$.
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State the double angle formula for cosine.
State the double angle formula for cosine.
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$\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$. One of the three forms of the double angle formula.
$\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$. One of the three forms of the double angle formula.
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What is the sine of $180^\text{o}$?
What is the sine of $180^\text{o}$?
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- At 180°, we're on the negative x-axis where y = 0.
- At 180°, we're on the negative x-axis where y = 0.
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What is the sine of a $90^\text{o}$ angle?
What is the sine of a $90^\text{o}$ angle?
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- At 90°, the opposite side equals the hypotenuse.
- At 90°, the opposite side equals the hypotenuse.
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Find the value of $\text{tan}^{-1}(1)$ in degrees.
Find the value of $\text{tan}^{-1}(1)$ in degrees.
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$45^\text{o}$. Inverse tangent of 1 gives the angle where $\tan(\theta) = 1$.
$45^\text{o}$. Inverse tangent of 1 gives the angle where $\tan(\theta) = 1$.
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What is the reciprocal of tangent?
What is the reciprocal of tangent?
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Cotangent. Reciprocal function: $\cot(\theta) = \frac{1}{\tan(\theta)}$.
Cotangent. Reciprocal function: $\cot(\theta) = \frac{1}{\tan(\theta)}$.
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What is the tangent of a $45^\text{o}$ angle?
What is the tangent of a $45^\text{o}$ angle?
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- In a 45-45-90 triangle, opposite and adjacent sides are equal.
- In a 45-45-90 triangle, opposite and adjacent sides are equal.
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What is the range of the cosine function?
What is the range of the cosine function?
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[-1, 1]. Cosine oscillates between -1 and 1 for all angles.
[-1, 1]. Cosine oscillates between -1 and 1 for all angles.
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Identify the phase shift in $y = \text{sin}(x - \frac{\text{\textpi}}{3})$.
Identify the phase shift in $y = \text{sin}(x - \frac{\text{\textpi}}{3})$.
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$\frac{\text{\textpi}}{3}$ to the right. Subtracting from $x$ shifts the graph to the right.
$\frac{\text{\textpi}}{3}$ to the right. Subtracting from $x$ shifts the graph to the right.
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State the formula for $\text{tan}(\theta)$ in terms of sine and cosine.
State the formula for $\text{tan}(\theta)$ in terms of sine and cosine.
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$\text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)}$. Definition of tangent as the ratio of sine to cosine.
$\text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)}$. Definition of tangent as the ratio of sine to cosine.
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