right triangle

Examples of right triangle in the following topics:

• How Trigonometric Functions Work

  • Trigonometric functions can be used to solve for missing side lengths in right triangles.
  • The hypotenuse is the side of the triangle opposite the right angle, and it is the longest.
  • The trigonometric functions are equal to ratios that relate certain side lengths of a  right triangle.  
  • Given a right triangle with acute angle of $34$ degrees and a hypotenuse length of $25$ feet, find the opposite side length.
  • The sides of a right triangle in relation to angle $t$.

• Right Triangles and the Pythagorean Theorem

  • A right triangle is a triangle in which one angle is a right angle.
  • The relation between the sides and angles of a right triangle is the basis for trigonometry.
  • It defines the relationship among the three sides of a right triangle.
  • Example 2:  A right triangle has side lengths $3$ cm and $4$ cm.  
  • Use the Pythagorean Theorem to find the length of a side of a right triangle

• Finding Angles From Ratios: Inverse Trigonometric Functions

  • The inverse trigonometric functions can be used to find the acute angle measurement of a right triangle.
  • Finding the missing acute angle when given two sides of a right triangle is just as simple.
  • 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:
  • $\displaystyle{ \begin{aligned} \sin{A^{\circ}} &= \frac {opposite}{hypotenuse} \\ \sin{A^{\circ}} &= \frac{12}{25} \\ A^{\circ} &= \sin^{-1}{\left( \frac{12}{25} \right)} \\ A^{\circ} &= \sin^{-1}{\left( 0.48 \right)} \\ A &=29^{\circ} \end{aligned} }$
  • Recognize the role of inverse trigonometric functions in solving problems about right triangles

• Sine, Cosine, and Tangent

  • The mnemonic SohCahToa can be used to solve for the length of a side of a right triangle.
  • Given a right triangle with an acute angle of $t$, the first three trigonometric functions are:
  • Given a right triangle with an acute angle of $62$ degrees and an adjacent side of $45$ feet, solve for the opposite side length.
  • The sides of a right triangle in relation to angle $t$.
  • Use the acronym SohCahToa to define Sine, Cosine, and Tangent in terms of right triangles

• Converting Between Polar and Cartesian Coordinates

  • Dropping a perpendicular from the point in the plane to the $x$-axis forms a right triangle, as illustrated in Figure below.
  • A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.
  • A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.

• The Law of Sines

  • The law of sines can be used to find unknown angles and sides in any triangle.
  • In previous sections, we have used trigonometry to find the measures of the angles and sides of right triangles.
  • We will now discuss the law of sines, which allows us to solve any triangle.
  • Note that any triangle that is not a right triangle is an oblique triangle.
  • In this triangle, $\alpha = 50\degree$, $\gamma = 30\degree$, and $a=10$.

• 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 .
  • Notice that the length between each point and the triangle's right angle is found by calculating the difference between the $y$-coordinates and $x$-coordinates, respectively.  
  • The distance formula includes the lengths of the legs of the triangle (normally labeled $a$ and $b$), with the expressions $(y_{2}-y_{1})$ and $(x_{2}-x_{1})$.
  • The distance formula between two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, shown as the hypotenuse of a right triangle

• The Law of Cosines

  • The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles.
  • Notice that if any angle $\theta$ in the triangle is a right angle (of measure 90°), then $\cos \theta = 0$, and the last term in the law of cosines cancels.
  • Thus, for right triangles, the law of cosines reduces to the Pythagorean theorem:
  • Find the length of the unknown side $b$ of the triangle below.
  • Substitute the values of $a$, $c$, and $\beta$ from the given triangle:

• Binomial Expansions and Pascal's Triangle

  • The rows of Pascal's triangle are numbered, starting with row $n = 0$ at the top.
  • A simple construction of the triangle proceeds in the following manner.
  • Notice that the entire right diagonal of Pascal's triangle corresponds to the coefficient of $y^n$ in these binomial expansions, while the next diagonal corresponds to the coefficient of $xy^{n−1}$ and so on.
  • It can be observed in the triangle that row 5 is $1, 5, 10, 10, 5, 1$.
  • Each number in the triangle is the sum of the two directly above it.

• Applications of Geometric Series

  • He determined that each green triangle has $\frac{1}{8}$ the area of the blue triangle, each yellow triangle has $\frac{1}{8}$ the area of a green triangle, and so forth.
  • $\displaystyle{1+2 \left ( \frac { 1 }{ 8 } \right ) +4 { \left ( \frac { 1 }{ 8 } \right ) }^{ 2 }+8{ \left ( \frac { 1 }{ 8 } \right ) }^{ 3 }+ \cdots}$
  • The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on.
  • Similarly, each triangle added in the second iteration has $\frac{1}{9}$ the area of the triangles added in the previous iteration, and so forth.
  • $\displaystyle{1+3 \left ( \frac { 1 }{ 9 } \right ) +12{ \left ( \frac { 1 }{ 9 } \right ) }^{ 2 }+48{ \left ( \frac { 1 }{ 9 } \right ) }^{ 3 }+ \cdots}$