hypotenuse

Examples of hypotenuse in the following topics:

• How Trigonometric Functions Work

  • The ratio of the sides would be the opposite side and the hypotenuse.  
  • Example 2: Given a right triangle with an acute angle of $83^{\circ}$ and a hypotenuse length of $300$ feet, find the hypotenuse length (round to the nearest tenth):
  • Looking at the figure, solve for the hypotenuse to the acute angle of $83$ degrees.
  • The ratio of the sides would be the adjacent side and the hypotenuse.  
  • Given a right triangle with an acute angle of $83$ degrees and a hypotenuse length of $300$ feet, find the hypotenuse length.

• Finding Angles From Ratios: Inverse Trigonometric Functions

  • Example 1:  For a right triangle with hypotenuse length $25~\mathrm{feet}$, and opposite side length $12~\mathrm{feet}$, find the acute angle to the nearest degree:
  • From angle $A$, the sides opposite and hypotenuse are given.  
  • $\displaystyle{ \begin{aligned} \sin{A^{\circ}} &= \frac {opposite}{hypotenuse} \\ \sin{A^{\circ}} &= \frac{12}{25} \\ A^{\circ} &= \sin^{-1}{\left( \frac{12}{25} \right)} \\ A^{\circ} &= \sin^{-1}{\left( 0.48 \right)} \\ A &=29^{\circ} \end{aligned} }$
  • Find the measure of angle $A$, when given the opposite side and hypotenuse.

• Sine, Cosine, and Tangent

  • Sine             $\displaystyle{ \sin{t} = \frac {opposite}{hypotenuse} }$
  • A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse (Soh), Cosine is adjacent over hypotenuse (Cah), Tangent is opposite over adjacent (Toa).”
  • Determine which trigonometric function to use when given the hypotenuse, and you need to solve for the opposite side.  
  • Remembering the mnemonic, "SohCahToa", the sides given are the hypotenuse and opposite or "h" and "o", which would use "S" or the sine trigonometric function.
  • $\displaystyle{ \begin{aligned} \sin{t} &= \frac {opposite}{hypotenuse} \\ \sin{ \left( 32^{\circ} \right) } & =\frac{x}{30} \\ 30\cdot \sin{ \left(32^{\circ}\right)} &=x \\ x &= 30\cdot \sin{ \left(32^{\circ}\right)}\\ x &= 30\cdot \left( 0.5299\dots \right) \\ x &= 15.9 ~\mathrm{feet} \end{aligned} }$

• Right Triangles and the Pythagorean Theorem

  • The side opposite the right angle is called the hypotenuse (side $c$ in the figure).
  • In this equation, $c$ represents the length of the hypotenuse and $a$ and $b$ the lengths of the triangle's other two sides.
  • Example 1:  A right triangle has a side length of $10$ feet, and a hypotenuse length of $20$ feet.  
  • Find the length of the hypotenuse.
  • The sum of the areas of the two squares on the legs ($a$ and $b$) is equal to the area of the square on the hypotenuse ($c$).  

• Arc Length and Speed

  • By the Pythagorean Theorem, each hypotenuse will have length $\sqrt{dx^2 + dy^2}$.
  • Adding up each tiny hypotenuse yields the arc length.
  • where the rate of change of the hypotenuse length depends on the rate of change of $x$ and $y$.
  • The length of a curve can be approximated by using a series of right triangles with the hypotenuses lying along the curve.

• Trigonometric Functions

  • The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
  • In our case $\sin A = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {a} {h}$ .
  • The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse: so called because it is the sine of the complementary or co-angle.
  • In our case $\cos A = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {b} {h}$.

• The Distance Formula and Midpoints of Segments

  • This formula is easily derived by constructing a right triangle with the hypotenuse connecting the two points ($c$) and two legs drawn from the each of the two points to intersect each other ($a$ and $b$), (see image below) and applying the Pythagorean theorem.
  • This theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides .
  • The distance formula between two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, shown as the hypotenuse of a right triangle
  • The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

• Secant and the Trigonometric Cofunctions

  • It can be described as the ratio of the length of the hypotenuse to the length of the adjacent side in a triangle.
  • $\displaystyle{ \begin{aligned} \sec x &= \frac{1}{\cos x} \\ \sec x &= \frac{\text{hypotenuse}}{\text{adjacent}} \end{aligned} }$
  • It can be described as the ratio of the length of the hypotenuse to the length of the opposite side in a triangle.
  • $\displaystyle{ \begin{aligned} \csc x &= \frac{1}{\sin x} \\ \csc x &= \frac{\text{hypotenuse}}{\text{opposite}} \end{aligned} }$
  • It can be described as the ratio of the length of the adjacent side to the length of the hypotenuse in a triangle.

• Scientific Applications of Quadratic Functions

  • This says that the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the two legs ($a$ and $b$) of the triangle.
  • Euclid used this diagram to explain how the sum of the squares of the triangle's smaller sides (pink and blue) sum to equal the area of the square of the hypotenuse.

• Components of a Vector

  • You should find you have a right triangle such that the original vector is the hypotenuse.