Precalculus Flashcards: Deriving The Triangle Area Formula

What this deck covers

This deck focuses on Deriving The Triangle Area Formula, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: A ramp rises 3 m over a 4 m run. What is the ramp length (hypotenuse)?

Answer: $5\text{ m}$. Use Pythagorean theorem: length $= \sqrt{3^2+4^2} = \sqrt{25} = 5$ m.

Flashcard 2: What is the complementary-angle relationship between sine and cosine in a right triangle?

Answer: $\sin(\theta)=\cos(90^\circ-\theta)$. Sine and cosine of complementary angles are equal.

Flashcard 3: Find $\tan(\theta)$ if opposite $=9$ and adjacent $=12$.

Answer: $\tan(\theta)=\frac{3}{4}$. Simplify $\frac{9}{12}$ to lowest terms: $\frac{3}{4}$.

Flashcard 4: State the reciprocal identity for $\csc(\theta)$ in terms of $\sin(\theta)$.

Answer: $\csc(\theta)=\frac{1}{\sin(\theta)}$. Cosecant is the reciprocal of sine.

Flashcard 5: Identify the missing leg: If $c=13$ and $a=5$, what is the other leg $b$?

Answer: $b=12$. Use Pythagorean theorem: $b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.

Flashcard 6: Find the horizontal distance $d$ if the height is $10$ and the angle of elevation is $45^\circ$.

Answer: $d=10$. With $\tan(45°)=1$, ground distance equals height.

Flashcard 7: State the reciprocal identity for $\sec(\theta)$ in terms of $\cos(\theta)$.

Answer: $\sec(\theta)=\frac{1}{\cos(\theta)}$. Secant is the reciprocal of cosine.

Flashcard 8: Find $\cos(\theta)$ if adjacent $=24$ and hypotenuse $=25$.

Answer: $\cos(\theta)=\frac{24}{25}$. Direct substitution into the cosine ratio formula.

Flashcard 9: What is the definition of cos(theta) in a right triangle?

Answer: $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$. Ratio of the side adjacent to angle $\theta$ over the hypotenuse.

Flashcard 10: Identify the missing side: If $a=9$ and $b=12$, what is the hypotenuse $c$?

Answer: $c=15$. Use Pythagorean theorem: $c=\sqrt{9^2+12^2}=\sqrt{81+144}=\sqrt{225}=15$.

Flashcard 11: Identify the angle of depression in a right-triangle model of looking down at an object.

Answer: Angle between horizontal and the line of sight downward. Measured from horizontal down to the line of sight.

Flashcard 12: Find the adjacent side: If hypotenuse $=20$ and $\cos(\theta)=\frac{4}{5}$, what is adjacent?

Answer: $16$. Multiply: adjacent $= \cos(\theta) \times$ hypotenuse $= \frac{4}{5} \times 20 = 16$.

Flashcard 13: Find the missing leg $b$ if the hypotenuse is $13$ and the other leg is $5$.

Answer: $b=12$. Use $b=\sqrt{c^2-a^2}=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.

Flashcard 14: State the reciprocal identity for $\cot(\theta)$ in terms of $\tan(\theta)$.

Answer: $\cot(\theta)=\frac{1}{\tan(\theta)}$. Cotangent is the reciprocal of tangent.

Flashcard 15: What is the definition of sin(theta) in a right triangle?

Answer: $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Ratio of the side opposite to angle $\theta$ over the hypotenuse.

Flashcard 16: State the complementary angle identity for sine and cosine in right triangles.

Answer: $\sin(\theta)=\cos(90^\circ-\theta)$. Sine of an angle equals cosine of its complement in a right triangle.

Flashcard 17: What is $\cos(\theta)$ in a right triangle in terms of adjacent and hypotenuse?

Answer: $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$. Ratio of the side next to angle $\theta$ over the longest side.

Flashcard 18: A 13 ft ladder reaches a 5 ft height on a wall. What is the distance from wall to ladder base?

Answer: $12\text{ ft}$. Use Pythagorean theorem: base $= \sqrt{13^2-5^2} = \sqrt{144} = 12$ ft.

Flashcard 19: Identify the angle of elevation in a right-triangle model of looking up at an object.

Answer: Angle between horizontal and the line of sight upward. Measured from horizontal up to the line of sight.

Flashcard 20: What is the definition of tan(theta) in a right triangle?

Answer: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Ratio of the side opposite to angle $\theta$ over the adjacent side.

Flashcard 21: State the Pythagorean Theorem for a right triangle with legs $a,b$ and hypotenuse $c$.

Answer: $a^2+b^2=c^2$. Relates the squares of the legs to the square of the hypotenuse in any right triangle.

Flashcard 22: What is $\tan(\theta)$ in a right triangle in terms of opposite and adjacent?

Answer: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Ratio of opposite over adjacent sides, or $\frac{\sin(\theta)}{\cos(\theta)}$.

Flashcard 23: State the complementary angle identity for tangent in right triangles.

Answer: $\tan(\theta)=\frac{1}{\tan(90^\circ-\theta)}$. Tangent of an angle equals the reciprocal of tangent of its complement.

Flashcard 24: Find the opposite side: If adjacent $=15$ and $\tan(\theta)=\frac{2}{3}$, what is opposite?

Answer: $10$. Multiply: opposite $= \tan(\theta) \times$ adjacent $= \frac{2}{3} \times 15 = 10$.

Flashcard 25: Find $\sin(\theta)$ if opposite $=7$ and hypotenuse $=25$.

Answer: $\sin(\theta)=\frac{7}{25}$. Direct substitution into the sine ratio formula.

Flashcard 26: Find the hypotenuse $c$ if the legs are $6$ and $8$.

Answer: $c=10$. Apply Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.

Flashcard 27: Find the angle theta if $\text{opp}=3$ and $\text{adj}=3$ in a right triangle.

Answer: $\theta=45^\circ$. Since $\tan(\theta)=\frac{3}{3}=1$, and $\tan(45°)=1$.

Flashcard 28: Find the height $h$ if the ground distance is $10$ and the angle of elevation is $45^\circ$.

Answer: $h=10$. With $\tan(45°)=1$, height equals ground distance.

Flashcard 29: Find the adjacent side if $h=10$ and $\theta=60^\circ$ with $\cos(\theta)=\frac{\text{adj}}{h}$.

Answer: $\text{adj}=5$. Since $\cos(60°)=0.5$, adjacent $=10 \times 0.5=5$.

Flashcard 30: State the complementary-angle relationship between tangent and cotangent in a right triangle.

Answer: $\tan(\theta)=\cot(90^\circ-\theta)$. Tangent and cotangent of complementary angles are equal.

Flashcard 31: Find the opposite side: If hypotenuse $=10$ and $\sin(\theta)=\frac{3}{5}$, what is opposite?

Answer: $6$. Multiply: opposite $= \sin(\theta) \times$ hypotenuse $= \frac{3}{5} \times 10 = 6$.

Flashcard 32: What is $\sin(\theta)$ in a right triangle in terms of opposite and hypotenuse?

Answer: $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Ratio of the side opposite to angle $\theta$ over the longest side.

Flashcard 33: Find the opposite side if $h=10$ and $\theta=30^\circ$ with $\sin(\theta)=\frac{\text{opp}}{h}$.

Answer: $\text{opp}=5$. Since $\sin(30°)=0.5$, opposite $=10 \times 0.5=5$.

Flashcard 34: A right triangle has legs $6$ and $8$. What is $\cos(\theta)$ for the angle adjacent to the $8$ side?

Answer: $\cos(\theta)=\frac{4}{5}$. Hypotenuse $= \sqrt{6^2+8^2} = 10$, so $\cos(\theta) = \frac{8}{10} = \frac{4}{5}$.