right triangle

(noun)

A $3$-sided shape where one angle has a value of $90$ degrees

Related Terms

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

  • Trigonometric Functions

    • Trigonometric functions are functions of an angle, relating the angles of a triangle to the lengths of the sides of a triangle.
    • They are used to relate the angles of a triangle to the lengths of the sides of a triangle.
    • Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle.
    • Note that this ratio does not depend on size of the particular right triangle chosen, as long as it contains the angle $A$, since all such triangles are similar.
    • If two right triangles have equal acute angles, they are similar, so their side lengths are proportional.

  • The Law of Sines

    • The law of sines can be used to find unknown angles and sides in any triangle.
    • Previous concepts explained how to use trigonometry to find the measures of the angles and sides of right triangles.
    • A right triangle contains a $90^{\circ}$ angle, while any other triangle is an oblique triangle.
    • $\displaystyle{ \begin{aligned} \frac{b \cdot \sin{\left(50^{\circ}\right)}}{10} &= \sin{\left(100^{\circ}\right)} \\ b \cdot \sin{\left(50^{\circ}\right)} &= 10 \cdot \sin{\left(100^{\circ}\right)} \\ b &= \frac{10 \cdot \sin{\left(100^{\circ}\right)}}{\sin{\left(50^{\circ}\right)}} \\ b &\approx 12.9 \end{aligned} }$
    • In this triangle, $\alpha = 50\degree$, $\gamma = 30\degree$, and $a=10$.

  • The Law of Cosines

    • The Law of Cosines is a more general form of the Pythagorean theorem, which holds only for right triangles.
    • Notice that if any angle $\theta$ in the triangle is a right angle (of measure $90^{\circ}$), 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:
    • From the unit circle, we find that $\displaystyle{\cos{\left(30^{\circ}\right)} = \frac{\sqrt{3}}{2}}$.
    • $\displaystyle{ \begin{aligned} b^2 &= 10^2 + 12^2 - 2\left(10\right)\left(12\right)\left(\frac{\sqrt{3}}{2}\right) \\ b^2 &= 100 + 144 - 240\frac{\sqrt{3}}{2} \\ b^2 &= 244 - 120\sqrt{3} \\ b &= \sqrt{244 - 120\sqrt{3}} \\ b &\approx 6.0 \end{aligned} }$

  • 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

    • $\displaystyle{ \begin{aligned} 0.7777 \cdots &= \frac { a }{ 1-r } \\ &= \frac { \frac{ 7 }{ 10 } }{ 1-\frac{ 1 }{ 10 } } \\ &= \frac{\left(\frac{7}{10}\right)}{\left(\frac{9}{10}\right)} \\ &= \left(\frac{7}{10}\right)\left(\frac{10}{9}\right)\\ &= \frac { 7 }{ 9 } \end{aligned} }$
    • He determined that each green triangle has $\displaystyle{\frac{1}{8}}$ the area of the blue triangle, each yellow triangle has $\displaystyle{\frac{1}{8}}$ 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.
    • $\displaystyle{1+3 \left ( \frac { 1 }{ 9 } \right ) +12{ \left ( \frac { 1 }{ 9 } \right ) }^{ 2 }+48{ \left ( \frac { 1 }{ 9 } \right ) }^{ 3 }+ \cdots}$

  • Impact of Changing Price on Producer Surplus

    • Conversely, if demand increases, and the demand curve shifts to the right, producer surplus increases.
    • At an initial demand represented by the "Demand (1)" curve, producer surplus is the blue triangle made of $P_1$, $A$, and $B$.
    • At an initial supply represented by the "Supply (1)" curve, producer surplus is the blue triangle made of $P_1$, $A$, and $C$.
    • If supply increases, represented by the "Supply (2)" curve, producer surplus is the larger gray triangle made of $P_2$, $B$, and $D$.
    • If the demand curve shifts out, producer surplus increases, as seen by size of the gray triangle.