Note 2.1.
It is usual to label the angles of a triangle with capital letters, and the side opposite each angle with the corresponding lower-case letter, as shown at right. We will follow this practice unless indicated otherwise.
Chapter 2: Trigonometric Ratios
Solve the inequality.
1. [latex]6-x > 3[/latex]
2. [latex]\dfrac{-3x}{4} \geq -6[/latex]
3. [latex]3x-7 \leq -10[/latex]
4. [latex]2x+9 > 4-3x[/latex]
If [latex]0> x[/latex], which of the following expressions are positive, and which are negative?
5. [latex]- x[/latex]
6. [latex]-(-x)[/latex]
7. [latex]|x|[/latex]
8. [latex]-|x|[/latex]
9. [latex]-|-x|[/latex]
10. [latex]x^{-1}[/latex]
Learning Objectives
From geometry, we know that the sum of the angles in a triangle is [latex]180°.[/latex] Are there any relationships between the angles of a triangle and its sides?
First of all, you have probably observed that the longest side in a triangle is always opposite the largest angle, and the shortest side is opposite the smallest angle, as illustrated below.
It is usual to label the angles of a triangle with capital letters, and the side opposite each angle with the corresponding lower-case letter, as shown at right. We will follow this practice unless indicated otherwise.
Example 2.2.
In [latex]\triangle FGH, \angle F=48^o,[/latex] and [latex]\angle G[/latex] is obtuse. Side [latex]f[/latex] is [latex]6[/latex] feet long. What can you conclude about the other sides?
Because [latex]\angle G[/latex] is greater than [latex]90^o\text{,}[/latex] we know that [latex]\angle F +\angle G[/latex] is greater than [latex]90^o + 48^o = 138^o\text{,}[/latex] so [latex]\angle F[/latex] is less than [latex]180^o-138^o = 42^o.[/latex] Thus, [latex]\angle H \lt \angle F \lt \angle G,[/latex] and consequently, [latex]h \lt f \lt g\text{.}[/latex] We can conclude that [latex]h \lt 6[/latex] feet long, and [latex]g \gt 6[/latex] feet long.
In isosceles triangle [latex]\triangle RST\text{,}[/latex] the vertex angle [latex]\angle S = 72^o\text{.}[/latex] Which side is longer, [latex]s[/latex] or [latex]t\text{?}[/latex]
[latex]s[/latex] is longer
It is also true that the sum of the lengths of any two sides of a triangle must be greater than the third side, or else the two sides will not meet to form a triangle. This fact is called the triangle inequality.
Triangle Inequality.
In any triangle, we must have that
[latex]\begin{equation*} p+q \gt r \end{equation*}[/latex]
where [latex]p, q, \text{and}~ r[/latex] are the lengths of the sides of the triangle.
We cannot use the triangle inequality to find the exact lengths of the sides of a triangle, but we can find the largest and smallest possible values for the length.
Example 2.4.
Two sides of a triangle have lengths [latex]7[/latex] inches and [latex]10[/latex] inches, as shown at right. What can you say about the length of the third side?
We let [latex]x[/latex] represent the length of the third side of the triangle. By looking at each side in turn, we can apply the triangle inequality three different ways to get
[latex]\begin{equation*} 7 \lt x+10, ~~~ 10 \lt x+7, ~~~ \text{and} ~~~ x \lt 10+7 \end{equation*}[/latex]
We solve each of these inequalities to find
[latex]\begin{equation*} -3 \lt x, ~~~ 3 \lt x, ~~~ \text{and} ~~~ x \lt 17 \end{equation*}[/latex]
We already know that [latex]x \gt -3[/latex] because [latex]x[/latex] must be positive, but the other two inequalities do give us new information. The third side must be greater than [latex]3[/latex] inches but less than [latex]17[/latex] inches long.
Can you make a triangle with three wooden sticks of lengths [latex]14[/latex] feet, [latex]26[/latex] feet, and [latex]10[/latex] feet? Sketch a picture, and explain why or why not.
No, [latex]10+14[/latex] is not greater than [latex]26[/latex].
In Chapter 1 we used the Pythagorean theorem to derive the distance formula. We can also use the Pythagorean theorem to find one side of a right triangle if we know the other two sides.
In a right triangle, if [latex]c[/latex] stands for the length of the hypotenuse, and the lengths of the two legs are denoted by [latex]a[/latex] and [latex]b\text{,}[/latex] then
[latex]\begin{equation*} a^2 + b^2 = c^2 \end{equation*}[/latex]
Example 2.6.
A [latex]25[/latex]-foot ladder is placed against a wall so that its foot is [latex]7[/latex] feet from the base of the wall. How far up the wall does the ladder reach?
We make a sketch of the situation, as shown below, and label any known dimensions. We’ll call the unknown height [latex]h\text{.}[/latex]
The ladder forms the hypotenuse of a right triangle, so we can apply the Pythagorean theorem, substituting [latex]25[/latex] for [latex]c\text{,}[/latex] [latex]7[/latex] for [latex]b\text{,}[/latex] and [latex]h[/latex] for [latex]a\text{.}[/latex]
[latex]\begin{align*} a^2 + b^2 = c^2\\ h^2 + 7^2 = 25^2 \end{align*}[/latex]
Now solve by extraction of roots:
[latex]\begin{align*} h^2 + 49 = 625 \ \ \text{Subtract 49 from both sides.}\\ h^2 = 576 \ \ \text{Extract roots.}\\ h = \pm \sqrt{576}\ \ \text{Simplify the radical.}\\ h = \pm 24 \end{align*}[/latex]
The height must be a positive number, so the solution [latex]-24[/latex] does not make sense for this problem. The ladder reaches [latex]24[/latex] feet up the wall.
A baseball diamond is a square whose sides are [latex]90[/latex] feet long. The catcher at home plate sees a runner on first trying to steal second base, and throws the ball to the second baseman. Find the straight-line distance from home plate to second base.
[latex]90\sqrt{2} \approx 127.3[/latex] feet
Keep in mind that the Pythagorean theorem is true only for right triangles, so the converse of the theorem is also true. In other words, if the sides of a triangle satisfy the relationship [latex]a^2 + b^2 = c^2\text{,}[/latex] then the triangle must be a right triangle. We can use this fact to test whether or not a given triangle has a right angle.
Example 2.9.
Delbert is paving a patio in his backyard and would like to know if the corner at [latex]C[/latex] is a right angle.
He measures [latex]20[/latex] cm along one side from the corner and [latex]48[/latex] cm along the other side, placing pegs [latex]P[/latex] and [latex]Q[/latex] at each position, as shown at right. The line joining those two pegs is [latex]52[/latex] cm long. Is the corner a right angle?
If is a right triangle, then its sides must satisfy [latex]p^2 + q^2 = c^2\text{.}[/latex] We find
[latex]\begin{align*} p^2 + q^2 = 20^2 + 48^2 = 400 + 2304 = 2704\\ c^2 = 52^2 = 2704 \end{align*}[/latex]
Yes, because [latex]p^2 + q^2 = c^2\text{,}[/latex] the corner at [latex]C[/latex] is a right angle.
The sides of a triangle measure 15 inches, 25 inches, and 30 inches long. Is the triangle a right triangle?
No
The Pythagorean theorem relates the sides of right triangles. However, for information about the sides of other triangles, the best we can do (without trigonometry!) is the triangle inequality. Nor does the Pythagorean theorem help us find the angles in a triangle. In the next section, we discover relationships between the angles and the sides of a right triangle.
The converse of a statement is a proposition obtained by reversing its terms. For example, the converse of the statement “If it is raining, then the streets are wet” is “If the streets are wet, then it is raining.” Note that the converse may or may not be true, even if the original statement is true.
Extraction of roots refers to the process of finding the value of a root (usually square, cube, or higher) of a number or expression, either by using a numerical method or by solving an equation.
An inequality is a mathematical statement that compares two values, expressions, or quantities using an inequality symbol (such as “>” for greater than, “>=” for greater than or equal to), indicating that one value is not equal to the other.