Deriving the Triangle Area Formula - Pre-Calculus
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Find the missing leg $b$ if the hypotenuse is $13$ and the other leg is $5$.
Find the missing leg $b$ if the hypotenuse is $13$ and the other leg is $5$.
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$b=12$. Use $b=\sqrt{c^2-a^2}=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.
$b=12$. Use $b=\sqrt{c^2-a^2}=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.
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Find the hypotenuse $c$ if the legs are $6$ and $8$.
Find the hypotenuse $c$ if the legs are $6$ and $8$.
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$c=10$. Apply Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.
$c=10$. Apply Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$.
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State the Pythagorean Theorem for a right triangle with legs $a,b$ and hypotenuse $c$.
State the Pythagorean Theorem for a right triangle with legs $a,b$ and hypotenuse $c$.
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$a^2+b^2=c^2$. Relates the squares of the legs to the square of the hypotenuse in any right triangle.
$a^2+b^2=c^2$. Relates the squares of the legs to the square of the hypotenuse in any right triangle.
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What is the definition of $sin(theta)$ in a right triangle?
What is the definition of $sin(theta)$ in a right triangle?
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$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Ratio of the side opposite to angle $\theta$ over the hypotenuse.
$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Ratio of the side opposite to angle $\theta$ over the hypotenuse.
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What is the definition of $cos(theta)$ in a right triangle?
What is the definition of $cos(theta)$ in a right triangle?
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$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$. Ratio of the side adjacent to angle $\theta$ over the hypotenuse.
$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$. Ratio of the side adjacent to angle $\theta$ over the hypotenuse.
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What is the definition of $tan(theta)$ in a right triangle?
What is the definition of $tan(theta)$ in a right triangle?
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$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Ratio of the side opposite to angle $\theta$ over the adjacent side.
$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Ratio of the side opposite to angle $\theta$ over the adjacent side.
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State the complementary angle identity for sine and cosine in right triangles.
State the complementary angle identity for sine and cosine in right triangles.
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$\sin(\theta)=\cos(90^\circ-\theta)$. Sine of an angle equals cosine of its complement in a right triangle.
$\sin(\theta)=\cos(90^\circ-\theta)$. Sine of an angle equals cosine of its complement in a right triangle.
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State the complementary angle identity for tangent in right triangles.
State the complementary angle identity for tangent in right triangles.
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$\tan(\theta)=\frac{1}{\tan(90^\circ-\theta)}$. Tangent of an angle equals the reciprocal of tangent of its complement.
$\tan(\theta)=\frac{1}{\tan(90^\circ-\theta)}$. Tangent of an angle equals the reciprocal of tangent of its complement.
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Identify the angle of elevation in a right-triangle model of looking up at an object.
Identify the angle of elevation in a right-triangle model of looking up at an object.
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Angle between horizontal and the line of sight upward. Measured from horizontal up to the line of sight.
Angle between horizontal and the line of sight upward. Measured from horizontal up to the line of sight.
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Identify the angle of depression in a right-triangle model of looking down at an object.
Identify the angle of depression in a right-triangle model of looking down at an object.
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Angle between horizontal and the line of sight downward. Measured from horizontal down to the line of sight.
Angle between horizontal and the line of sight downward. Measured from horizontal down to the line of sight.
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Find the opposite side if $h=10$ and $\theta=30^\circ$ with $\sin(\theta)=\frac{\text{opp}}{h}$.
Find the opposite side if $h=10$ and $\theta=30^\circ$ with $\sin(\theta)=\frac{\text{opp}}{h}$.
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$\text{opp}=5$. Since $\sin(30°)=0.5$, opposite $=10 \times 0.5=5$.
$\text{opp}=5$. Since $\sin(30°)=0.5$, opposite $=10 \times 0.5=5$.
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Find the adjacent side if $h=10$ and $\theta=60^\circ$ with $\cos(\theta)=\frac{\text{adj}}{h}$.
Find the adjacent side if $h=10$ and $\theta=60^\circ$ with $\cos(\theta)=\frac{\text{adj}}{h}$.
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$\text{adj}=5$. Since $\cos(60°)=0.5$, adjacent $=10 \times 0.5=5$.
$\text{adj}=5$. Since $\cos(60°)=0.5$, adjacent $=10 \times 0.5=5$.
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Find the height $h$ if the ground distance is $10$ and the angle of elevation is $45^\circ$.
Find the height $h$ if the ground distance is $10$ and the angle of elevation is $45^\circ$.
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$h=10$. With $\tan(45°)=1$, height equals ground distance.
$h=10$. With $\tan(45°)=1$, height equals ground distance.
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Find the horizontal distance $d$ if the height is $10$ and the angle of elevation is $45^\circ$.
Find the horizontal distance $d$ if the height is $10$ and the angle of elevation is $45^\circ$.
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$d=10$. With $\tan(45°)=1$, ground distance equals height.
$d=10$. With $\tan(45°)=1$, ground distance equals height.
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Find the angle $theta$ if $\text{opp}=3$ and $\text{adj}=3$ in a right triangle.
Find the angle $theta$ if $\text{opp}=3$ and $\text{adj}=3$ in a right triangle.
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$\theta=45^\circ$. Since $\tan(\theta)=\frac{3}{3}=1$, and $\tan(45°)=1$.
$\theta=45^\circ$. Since $\tan(\theta)=\frac{3}{3}=1$, and $\tan(45°)=1$.
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State the reciprocal identity for $\csc(\theta)$ in terms of $\sin(\theta)$.
State the reciprocal identity for $\csc(\theta)$ in terms of $\sin(\theta)$.
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$\csc(\theta)=\frac{1}{\sin(\theta)}$. Cosecant is the reciprocal of sine.
$\csc(\theta)=\frac{1}{\sin(\theta)}$. Cosecant is the reciprocal of sine.
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Find the adjacent side: If hypotenuse $=20$ and $\cos(\theta)=\frac{4}{5}$, what is adjacent?
Find the adjacent side: If hypotenuse $=20$ and $\cos(\theta)=\frac{4}{5}$, what is adjacent?
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$16$. Multiply: adjacent $= \cos(\theta) \times$ hypotenuse $= \frac{4}{5} \times 20 = 16$.
$16$. Multiply: adjacent $= \cos(\theta) \times$ hypotenuse $= \frac{4}{5} \times 20 = 16$.
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What is $\sin(\theta)$ in a right triangle in terms of opposite and hypotenuse?
What is $\sin(\theta)$ in a right triangle in terms of opposite and hypotenuse?
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$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Ratio of the side opposite to angle $\theta$ over the longest side.
$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Ratio of the side opposite to angle $\theta$ over the longest side.
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What is $\cos(\theta)$ in a right triangle in terms of adjacent and hypotenuse?
What is $\cos(\theta)$ in a right triangle in terms of adjacent and hypotenuse?
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$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$. Ratio of the side next to angle $\theta$ over the longest side.
$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$. Ratio of the side next to angle $\theta$ over the longest side.
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What is $\tan(\theta)$ in a right triangle in terms of opposite and adjacent?
What is $\tan(\theta)$ in a right triangle in terms of opposite and adjacent?
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$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Ratio of opposite over adjacent sides, or $\frac{\sin(\theta)}{\cos(\theta)}$.
$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Ratio of opposite over adjacent sides, or $\frac{\sin(\theta)}{\cos(\theta)}$.
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State the reciprocal identity for $\sec(\theta)$ in terms of $\cos(\theta)$.
State the reciprocal identity for $\sec(\theta)$ in terms of $\cos(\theta)$.
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$\sec(\theta)=\frac{1}{\cos(\theta)}$. Secant is the reciprocal of cosine.
$\sec(\theta)=\frac{1}{\cos(\theta)}$. Secant is the reciprocal of cosine.
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State the reciprocal identity for $\cot(\theta)$ in terms of $\tan(\theta)$.
State the reciprocal identity for $\cot(\theta)$ in terms of $\tan(\theta)$.
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$\cot(\theta)=\frac{1}{\tan(\theta)}$. Cotangent is the reciprocal of tangent.
$\cot(\theta)=\frac{1}{\tan(\theta)}$. Cotangent is the reciprocal of tangent.
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What is the complementary-angle relationship between sine and cosine in a right triangle?
What is the complementary-angle relationship between sine and cosine in a right triangle?
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$\sin(\theta)=\cos(90^\circ-\theta)$. Sine and cosine of complementary angles are equal.
$\sin(\theta)=\cos(90^\circ-\theta)$. Sine and cosine of complementary angles are equal.
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State the complementary-angle relationship between tangent and cotangent in a right triangle.
State the complementary-angle relationship between tangent and cotangent in a right triangle.
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$\tan(\theta)=\cot(90^\circ-\theta)$. Tangent and cotangent of complementary angles are equal.
$\tan(\theta)=\cot(90^\circ-\theta)$. Tangent and cotangent of complementary angles are equal.
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Identify the missing side: If $a=9$ and $b=12$, what is the hypotenuse $c$?
Identify the missing side: If $a=9$ and $b=12$, what is the hypotenuse $c$?
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$c=15$. Use Pythagorean theorem: $c=\sqrt{9^2+12^2}=\sqrt{81+144}=\sqrt{225}=15$.
$c=15$. Use Pythagorean theorem: $c=\sqrt{9^2+12^2}=\sqrt{81+144}=\sqrt{225}=15$.
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Identify the missing leg: If $c=13$ and $a=5$, what is the other leg $b$?
Identify the missing leg: If $c=13$ and $a=5$, what is the other leg $b$?
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$b=12$. Use Pythagorean theorem: $b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.
$b=12$. Use Pythagorean theorem: $b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$.
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Find $\sin(\theta)$ if opposite $=7$ and hypotenuse $=25$.
Find $\sin(\theta)$ if opposite $=7$ and hypotenuse $=25$.
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$\sin(\theta)=\frac{7}{25}$. Direct substitution into the sine ratio formula.
$\sin(\theta)=\frac{7}{25}$. Direct substitution into the sine ratio formula.
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Find $\cos(\theta)$ if adjacent $=24$ and hypotenuse $=25$.
Find $\cos(\theta)$ if adjacent $=24$ and hypotenuse $=25$.
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$\cos(\theta)=\frac{24}{25}$. Direct substitution into the cosine ratio formula.
$\cos(\theta)=\frac{24}{25}$. Direct substitution into the cosine ratio formula.
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Find $\tan(\theta)$ if opposite $=9$ and adjacent $=12$.
Find $\tan(\theta)$ if opposite $=9$ and adjacent $=12$.
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$\tan(\theta)=\frac{3}{4}$. Simplify $\frac{9}{12}$ to lowest terms: $\frac{3}{4}$.
$\tan(\theta)=\frac{3}{4}$. Simplify $\frac{9}{12}$ to lowest terms: $\frac{3}{4}$.
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Find the opposite side: If hypotenuse $=10$ and $\sin(\theta)=\frac{3}{5}$, what is opposite?
Find the opposite side: If hypotenuse $=10$ and $\sin(\theta)=\frac{3}{5}$, what is opposite?
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$6$. Multiply: opposite $= \sin(\theta) \times$ hypotenuse $= \frac{3}{5} \times 10 = 6$.
$6$. Multiply: opposite $= \sin(\theta) \times$ hypotenuse $= \frac{3}{5} \times 10 = 6$.
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