Trigonometry - SAT Math

Cosecant (csc). Since $\sin\theta = \frac{1}{\csc\theta}$, cosecant is sine's reciprocal.

$\frac{\text{sqrt}(2)}{2}$. Standard value for 45° in the unit circle.

$\text{tan} \theta = \frac{\text{opp}}{\text{adj}}$. Tangent equals opposite side divided by adjacent side.

1. At 90°, the x-coordinate on the unit circle is 0.

$\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$. Formula for sine of double angle.

$\frac{1}{\sqrt{3}}$. Using the 30-60-90 triangle ratios.

$2\pi$. Sine completes one cycle every $2\pi$ radians.

$\text{tan}(2\theta) = \frac{2\text{tan}(\theta)}{1-\text{tan}^2(\theta)}$. Formula for tangent of double angle.

$\frac{1}{2}$. Standard value for 60° in the unit circle.

1. At 0°, the adjacent side equals the hypotenuse.

$\text{cos}(60^\text{o}) = \frac{1}{2}$. Using $\sin(90° - 60°) = \cos 60°$.

$\pi$ radians. Use the conversion factor $\frac{\pi}{180}$.

1. At 0°, the x-coordinate on the unit circle is 1.

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$. Formula for tangent of sum of two angles.

$\text{cos}(2\theta) = -\frac{1}{2}$. Using double angle formula with given values.

$\text{sin}^2 \theta + \text{cos}^2 \theta = 1$. Fundamental identity derived from the Pythagorean theorem.

1. Since $\tan 45° = 1$, cotangent is the reciprocal.

$\frac{\text{sqrt}(3)}{2}$. Standard value for 30° in the unit circle.

$\text{sin}^2\theta + \text{cos}^2\theta = 1$. Fundamental trigonometric identity derived from the unit circle.

$\text{sin}(30^\text{o}) = \frac{1}{2}$. Using $\cos(90° - 30°) = \sin 30°$.

$\text{tan } 60^\text{o} = \frac{\text{sin } 60^\text{o}}{\text{cos } 60^\text{o}} = \frac{\frac{\text{sqrt}(3)}{2}}{\frac{1}{2}} = \text{sqrt}(3)$. Using the definition $\tan = \frac{\sin}{\cos}$.

Cosecant (csc). Cosecant is defined as $\frac{1}{\sin}$.

$\sin(2\theta) = \sqrt{3}$. Using double angle formula with given values.

$\text{sin}(A + B) = \text{sin}A \text{cos}B + \text{cos}A \text{sin}B$. Formula for sine of sum of two angles.

1. At 0°, the point on the unit circle is (1,0), so cosine equals 1.

$\frac{1}{2}ab \sin C$. Uses two sides and the included angle between them.

$\cos(\frac{\theta}{2}) = \sqrt{\frac{1 + \cos(\theta)}{2}}$. Formula for cosine of half angle.

$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$. Formula for cosine of double angle.

$\text{sin}(\frac{\theta}{2}) = \text{sqrt}(\frac{1 - \text{cos}(\theta)}{2})$. Formula for sine of half angle.

Secant. Secant function is defined as $\sec \theta = \frac{1}{\cos \theta}$.

1. At 90°, the y-coordinate on the unit circle is 1.

$\frac{1}{2}$. Standard value for 30° in the unit circle.

1. Secant is the reciprocal of cosine; $\cos 60° = \frac{1}{2}$.

1. At 90°, the opposite side equals the hypotenuse in a right triangle.

$\text{cos}(A + B) = \text{cos}A \text{cos}B - \text{sin}A \text{sin}B$. Formula for cosine of sum of two angles.

$\text{sin}^2 \theta + \text{cos}^2 \theta = 1$. Fundamental identity derived from the unit circle.

$\text{π}$. Tangent completes one cycle every $\pi$ radians.

$\text{cos}(90^\text{o} - \theta) = \text{sin}(\theta)$. Complementary angle relationship for cosine.

$\frac{\text{sqrt}(2)}{2}$. Standard value for 45° in the unit circle.

$\frac{1}{2}$. Square of $\frac{\sqrt{2}}{2}$ equals $\frac{1}{2}$.

Cotangent (cot). Cotangent is defined as $\frac{1}{\tan}$.

1. Square of $\tan 45° = 1$ is 1.

$\tan(\frac{\theta}{2}) = \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}$. Formula for tangent of half angle.

$\frac{\text{√}2}{2}$. In a 45-45-90 triangle, opposite and adjacent sides are equal.

$2\pi$. Cosine completes one cycle every $2\pi$ radians.

[-1, 1]. Cosine values are bounded between -1 and 1.

$\text{sin}(90^\text{o} - \theta) = \text{cos}(\theta)$. Complementary angle relationship for sine.

$\frac{\sqrt{3}}{2}$. Standard value for 60° in the unit circle.

$\frac{1}{2}$. Square of $\frac{\sqrt{2}}{2}$ equals $\frac{1}{2}$.

[-1, 1]. Sine values are bounded between -1 and 1.

$\frac{1}{2}$. In a 30-60-90 triangle, the shortest side is half the hypotenuse.

Secant (sec). Secant is defined as $\frac{1}{\cos}$.

1. Secant is the reciprocal of cosine; $\cos 60° = \frac{1}{2}$.

$\frac{1}{2}ab \sin C$. Uses two sides and the included angle between them.

$\frac{1}{2}$. In a 30-60-90 triangle, the shortest side is half the hypotenuse.

1. Since $\tan 45° = 1$, cotangent is the reciprocal.

Secant. Secant function is defined as $\sec \theta = \frac{1}{\cos \theta}$.

$\text{π}$ radians. Use the conversion factor $\frac{\pi}{180}$.

$\text{tan} \theta = \frac{\text{opp}}{\text{adj}}$. Tangent equals opposite side divided by adjacent side.

1. At 0°, the adjacent side equals the hypotenuse.

$\text{sin}(90^\text{o} - \theta) = \text{cos}(\theta)$. Complementary angle relationship for sine.

$\text{tan } 60^\text{o} = \frac{\text{sin } 60^\text{o}}{\text{cos } 60^\text{o}} = \frac{\frac{\text{sqrt}(3)}{2}}{\frac{1}{2}} = \text{sqrt}(3)$. Using the definition $\tan = \frac{\sin}{\cos}$.

$\frac{\sqrt{3}}{2}$. Standard value for 60° in the unit circle.

$\frac{1}{\text{sqrt}(3)}$. Using the 30-60-90 triangle ratios.

$\frac{1}{2}$. Square of $\frac{\sqrt{2}}{2}$ equals $\frac{1}{2}$.

$\text{cos}(90^\text{o} - \theta) = \text{sin}(\theta)$. Complementary angle relationship for cosine.

1. Square of $\tan 45° = 1$ is 1.

$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$. Formula for cosine of double angle.

$\text{sin}(30^\text{o}) = \frac{1}{2}$. Using $\cos(90° - 30°) = \sin 30°$.

$\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}}$. Formula for sine of half angle.

$\cos(\frac{\theta}{2}) = \sqrt{\frac{1 + \cos(\theta)}{2}}$. Formula for cosine of half angle.

$\text{tan}(\frac{\theta}{2}) = \text{sqrt}(\frac{1 - \text{cos}(\theta)}{1 + \text{cos}(\theta)})$. Formula for tangent of half angle.

$2\text{π}$. Sine completes one cycle every $2\pi$ radians.

$2\text{π}$. Cosine completes one cycle every $2\pi$ radians.

$\text{π}$. Tangent completes one cycle every $\pi$ radians.

$\sin(2\theta) = \sqrt{3}$. Using double angle formula with given values.

$\cos(2\theta) = -\frac{1}{2}$. Using double angle formula with given values.

[-1, 1]. Sine values are bounded between -1 and 1.

[-1, 1]. Cosine values are bounded between -1 and 1.

$\text{sin}^2 \theta + \text{cos}^2 \theta = 1$. Fundamental identity derived from the unit circle.

Secant (sec). Secant is defined as $\frac{1}{\cos}$.

$\text{sin}(A + B) = \text{sin}A \text{cos}B + \text{cos}A \text{sin}B$. Formula for sine of sum of two angles.

$\frac{\text{sqrt}(3)}{2}$. Standard value for 30° in the unit circle.

$\frac{1}{2}$. Square of $\frac{\sqrt{2}}{2}$ equals $\frac{1}{2}$.

$\text{tan}(2\theta) = \frac{2\text{tan}(\theta)}{1-\text{tan}^2(\theta)}$. Formula for tangent of double angle.

$\text{cos}(60^\text{o}) = \frac{1}{2}$. Using $\sin(90° - 60°) = \cos 60°$.

$\frac{1}{2}$. Standard value for 60° in the unit circle.

1. At 0°, the x-coordinate on the unit circle is 1.

$\text{tan}(A + B) = \frac{\text{tan}A + \text{tan}B}{1 - \text{tan}A \text{tan}B}$. Formula for tangent of sum of two angles.

1. At 90°, the x-coordinate on the unit circle is 0.

$\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$. Formula for sine of double angle.

$\frac{1}{2}$. Standard value for 30° in the unit circle.

1. In a 45-45-90 triangle, opposite equals adjacent.

1. At 90°, the y-coordinate on the unit circle is 1.

Cosecant (csc). Cosecant is defined as $\frac{1}{\sin}$.

Cotangent (cot). Cotangent is defined as $\frac{1}{\tan}$.

$\frac{\text{sqrt}(2)}{2}$. Standard value for 45° in the unit circle.

$\frac{\text{sqrt}(2)}{2}$. Standard value for 45° in the unit circle.

$\text{cos}(A + B) = \text{cos}A \text{cos}B - \text{sin}A \text{sin}B$. Formula for cosine of sum of two angles.

$\text{sin}^2\theta + \text{cos}^2\theta = 1$. Fundamental trigonometric identity derived from the unit circle.

1. At 0°, the point on the unit circle is (1,0), so cosine equals 1.

Cosecant (csc). Since $\sin\theta = \frac{1}{\csc\theta}$, cosecant is sine's reciprocal.

$\frac{\text{√}2}{2}$. In a 45-45-90 triangle, opposite and adjacent sides are equal.