Properties of Right Triangles - SAT Math

$\text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Tangent is the ratio of opposite side to adjacent side.

$\frac{1}{2} \times \text{base} \times \text{height}$. Uses the two perpendicular sides as base and height.

30 degrees. Complementary angles in a right triangle sum to 90 degrees.

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

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

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

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

$\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. Sine relates the side opposite to the angle with the hypotenuse.

$\text{π}$ 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.

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

$\text{tan } A = \frac{4}{3}$. Tangent equals sine divided by cosine.

Cosecant. Cosecant is the reciprocal of sine function.

$\text{sin}(90^\text{o} - \theta) = \text{cos}(\theta)$. 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 $\tan = \frac{\sin}{\cos}$.

Secant. Reciprocal trigonometric function of cosine.

$\frac{\text{opposite}}{\text{hypotenuse}}$. Ratio of side opposite angle to longest side.

$\frac{\text{adjacent}}{\text{hypotenuse}}$. Ratio of side adjacent to angle to hypotenuse.

$\frac{\text{opposite}}{\text{adjacent}}$. Ratio of opposite side to adjacent side.

$b = 12$. Using $9^2 + b^2 = 15^2$, so $b^2 = 144$.

$c = 5$. Using $3^2 + 4^2 = 9 + 16 = 25$, so $c = 5$.

1. Cosine of 0 degrees equals 1.

Cotangent. Reciprocal trigonometric function of tangent.

90 degrees. Right angle is always opposite the hypotenuse.

Each angle is $45^\text{o}$. Isosceles right triangle has two equal acute angles.

$\frac{1}{2} \times b \times h$. Standard area formula for triangles using base and height.

$\frac{1}{2}$. Standard trigonometric value for 30-degree angle.

$7\text{√}2$. Hypotenuse equals leg times $\sqrt{2}$ in 45-45-90 triangle.

$\frac{1}{2}$. Standard trigonometric value for 60-degree angle.

$5$. Both legs are equal in isosceles right triangle.

$3\text{√}3$. Longer leg equals shorter leg times $\sqrt{3}$.

Cotangent. Cotangent is the reciprocal of tangent function.

Cosecant. Reciprocal trigonometric function of sine.

$a = 8$. Using $a^2 + 6^2 = 10^2$, so $a^2 = 64$.

1. Tangent equals 1 when opposite equals adjacent.

$\text{sin}(2\theta) = 2 \text{sin}(\theta) \text{cos}(\theta)$. 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.

$\frac{\text{sqrt}(3)}{2}$. 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.

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

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

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

$\text{cos}(\frac{\theta}{2}) = \text{sqrt}(\frac{1 + \text{cos}(\theta)}{2})$. 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.

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

$\text{cos}(2\theta) = -\frac{1}{2}$. 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.

$\text{cos } A = 0.8$. Using $\sin^2 A + \cos^2 A = 1$ identity.

Sum is $180^\text{o}$. All triangle angles sum to 180 degrees.

$1:1:\text{√}2$. Special ratio for isosceles right triangle.

1. Cosecant of 90 degrees equals 1 divided by sin 90.

$5$. Shorter leg is half the hypotenuse in 30-60-90 triangle.

$1:\text{√}3:2$. Standard side ratio for 30-60-90 special right triangle.

Secant. Secant is the reciprocal of cosine function.

$c = 15$. Using $9^2 + 12^2 = 81 + 144 = 225$.

$45^\text{o}$. Angle whose tangent equals 1 is 45 degrees.

1. Secant of 0 degrees equals 1 divided by cos 0.

1. Cotangent of 45 degrees equals cos 45 divided by sin 45.

1. Sine of 90 degrees equals 1.

$c = 13$. Using $5^2 + 12^2 = 25 + 144 = 169$.

Pythagorean Theorem. Fundamental relationship between sides in right triangles.

$a = 6$. Using $a^2 + 8^2 = 10^2$, so $a^2 = 36$.

$c = \sqrt{a^2 + b^2}$. Square root of sum of squared legs gives hypotenuse.

$\tan(\theta) = \frac{5}{12}$. Tangent equals opposite divided by adjacent: $\frac{5}{12}$.

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.

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

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