oblique triangle

(noun)

A three-sided shape that does not contain a $90^{\circ}$ angle.

Examples of oblique triangle in the following topics:

  • The Law of Sines

    • The law of sines can be used to find unknown angles and sides in any triangle.
    • A right triangle contains a $90^{\circ}$ angle, while any other triangle is an oblique triangle.
    • Solving an oblique triangle means finding the measurements of all three angles and all three sides.
    • To solve an oblique triangle, use any pair of applicable ratios from the law of sines formula.
    • The sides of this oblique triangle are labeled a, b, and c, and the angles are labeled $\alpha$, $\beta$, and $\gamma$.

  • The Law of Cosines

    • The Law of Cosines defines the relationship among angle measurements and side lengths in oblique triangles.
    • Thus, for right triangles, the Law of Cosines reduces to the Pythagorean theorem:
    • Substitute the values of $a$, $c$, and $\beta$ from the given triangle:
    • An oblique triangle, with angles $\alpha$, $\beta$, and $\gamma$, and opposite corresponding sides $a$, $b$, and $c$.
    • This oblique triangle has known side lengths $a=10$ and $c=12$, and known angle $\beta = 30^{\circ}$.

  • Right Triangles and the Pythagorean Theorem

    • A right triangle is a triangle in which one angle is a right angle.
    • If the lengths of all three sides of a right triangle are whole numbers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple.
    • 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

  • How Trigonometric Functions Work

    • Trigonometric functions can be used to solve for missing side lengths in right triangles.
    • We can define the trigonometric functions in terms an angle $t$ and the lengths of the sides of the triangle.
    • 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.  
    • The sides of a right triangle in relation to angle $t$.

  • 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:
    • After sketching a picture of the problem, we have the triangle shown.
    • 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

  • 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:
    • Recognize the role of inverse trigonometric functions in solving problems about right triangles

  • Applications of Geometric Series

    • 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.
    • 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 $\displaystyle{\frac{1}{9}}$ the area of the triangles added in the previous iteration, and so forth.
    • The first term of this series represents the area of the first triangle, the second term the total area of the three triangles added in the second iteration, the third term the total area of the twelve triangles added in the third iteration, and so forth.
    • Each iteration adds a set of triangles to the outside of the shape.

  • Asymptotes

    • There are three kinds of asymptotes: horizontal, vertical and oblique.
    • An asymptote that is neither horizontal or vertical is an oblique (or slant) asymptote.
    • A rational function has at most one horizontal or oblique asymptote, and possibly many vertical asymptotes.
    • The graph of a function with a horizontal ($y=0$), vertical ($x=0$), and oblique asymptote (blue line).
    • Explain when the asymptote of a rational function will be horizontal, oblique, or vertical

  • 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 the coefficients are the numbers in row two of Pascal's triangle: $1,2,1$.
    • 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.

  • 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 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