How many triangles are in the figure? How many of those triangles are right triangles?
Source: Brilliant.org |
Trigonometric (trig) ratios can be used to solve for missing side lengths and/or angle measures of right triangles. There are six trig ratios that will be introduced, but only three of these ratios are actually used when solving for missing side lengths or angle measures of a right triangle.
A trig ratio (fraction) is a ratio of side lengths in relation to a given acute angle. Each acute angle has a specific ratio. For example, every angle that measures \(30^{\circ}\) has a ratio of sides equal to \(\frac{1}{2}\). Regardless of the side lengths, the ratio will be same because the triangles are similar.
Before we define the trig ratios, let’s look at a diagram. Let \(\theta\) (theta) be an acute angle of the right triangle.
A trig ratio (fraction) is a ratio of side lengths in relation to a given acute angle. Each acute angle has a specific ratio. For example, every angle that measures \(30^{\circ}\) has a ratio of sides equal to \(\frac{1}{2}\). Regardless of the side lengths, the ratio will be same because the triangles are similar.
Before we define the trig ratios, let’s look at a diagram. Let \(\theta\) (theta) be an acute angle of the right triangle.
Side \(y\) is opposite angle \(\theta\) because it is across from it (an opposite side will never intersect the vertex of the acute angle). Side \(x\) is adjacent to angle \(\theta\) because it is next to it (an adjacent side will intersect the vertex of the acute angle). Side \(r\) is the hypotenuse because it is opposite the right angle and is the longest side of the triangle. |
The six trig ratios are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec) and cotangent (cot).
Let \(\theta\) be an acute angle of a right triangle. The six trigonometric functions of \(\theta\) are defined as:
\(sin\theta=\large\frac{opposite}{hypotenuse}=\frac{y}{r}\) \(csc\theta=\large\frac{hypotenuse}{opposite}=\frac{r}{y}\)
\(\cos\theta=\large\frac{adjacent}{hypotenuse}=\frac{x}{r}\) \(\sec\theta=\large\frac{hypotenuse}{adjacent}=\frac{r}{x}\)
\(\tan\theta=\large\frac{opposite}{adjacent}=\frac{y}{x}\) \(\cot\theta=\large\frac{adjacent}{opposite}=\frac{x}{y}\)
Let \(\theta\) be an acute angle of a right triangle. The six trigonometric functions of \(\theta\) are defined as:
\(sin\theta=\large\frac{opposite}{hypotenuse}=\frac{y}{r}\) \(csc\theta=\large\frac{hypotenuse}{opposite}=\frac{r}{y}\)
\(\cos\theta=\large\frac{adjacent}{hypotenuse}=\frac{x}{r}\) \(\sec\theta=\large\frac{hypotenuse}{adjacent}=\frac{r}{x}\)
\(\tan\theta=\large\frac{opposite}{adjacent}=\frac{y}{x}\) \(\cot\theta=\large\frac{adjacent}{opposite}=\frac{x}{y}\)
There are some other useful equivalent expressions, called trigonometric identities, which can be useful. They are as follows:
Quotient Identities: Reciprocal Identities:
\(\tan\theta=\large\frac{\sin\theta}{\cos\theta}\) \(\csc\theta=\large\frac{1}{\sin\theta}\)
\(\cot\theta=\large\frac{\cos\theta}{\sin\theta}\) \(\sec\theta=\large\frac{1}{\cos\theta}\)
\(\cot\theta=\large\frac{1}{\tan\theta}\)
Example 1: Find the six trigonometric ratios of angle .
\(\sin\theta=\large\frac{5}{13}\) \(\csc\theta=\large\frac{13}{5}\)
\(\cos\theta=\large\frac{12}{13}\) \(\sec\theta=\large\frac{13}{12}\)
\(\tan\theta=\large\frac{5}{12}\) \(\cot\theta=\large\frac{12}{5}\)
Example 2: Let \(\theta\) be an acute angle of a right triangle. Find the values of the other five trigonometric functions of \(\theta\) given that \(\sec\theta=\frac{25}{7}\).
Now that we have defined our trig functions, we can use them to help us solve for missing sides or missing angles in right triangles. Here are a few things to keep in mind:
- The calculator must be in degree mode (Press MODE, highlight DEGREE then press ENTER).
- To solve for angles, you use inverse trig functions: (\(\sin^{-1},\cos^{-1},\tan^{-1}\)). The inverse functions are found by pressing 2nd then either sin, cos, or tan.
- Angles are represented by capital letters and sides by lower case letters.
- The side and the angle opposite the side share the same letter label.
- Solving a triangle means find all missing side lengths and angle measures.
Steps for solving for a missing side:
1. Draw and label the diagram.
2. In relation to the given angle, determine which side you know and which side you are solving for (opposite the
angle, ajacent to the angle or the hypotenuse).
3. Choose the correct trig ratio based on the sides.
4. Set up the trig equation and solve.
Example 3: Solve for side \(b\) in the triangle below:
Steps for solving for an angle:
1. Draw and label the diagram.
2. In relation to the angle you are solving for, identify the two sides you are given as either opposite the angle, adjacent
to the angle or the hypotenuse.
3. Choose the correct trig ratio based on the sides.
4. Set up the trig equation and solve.
Example 4: In \(\triangle ABC, \ AC = 13, \ BC = 15\ \text{and}\ \ C = 90^{\circ}\). Find the measure of angle \(A\) to the nearest thousandth.
Angle of Elevation and Angle of Depression
Drag the orange dot up and down (you can move the dot below the horizontal line too). Notice the angle at which the dude sees the orange dot. He's always looking out, horizontally, and up forming and angle of elevation or he's looking out, horizontally, and down, forming an angle of depression. An angle of elevation is the angle formed by the line of sight to the object (the hypotenuse of the right triangle) and a horizontal line (horizontal leg of the right triangle). An angle of depression (or descent) is the angle formed by the line of sight to the object (the hypotenuse of the right triangle) and a horizontal line (horizontal leg of the right triangle). |
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Another way to represent angle of elevation and angle of depression in the same picture is with a rectangle. The angle of elevation in this image is from the dude looking up towards the orange dot. The angle of depression in this image is formed by looking from the orange dot horizontally and down towards the dude.
Drag the orange dot. What do you notice about the angle of elevation and the angle of depression? Quick Check The angle of elevation is congruent to the angle of depression. Do you know why? Quick Check Solution |
Example 5: The angle of elevation from the base of the Adrena Zipline in West Virginia to the top of the zipline is \(7^{\circ}\). If the zipline is \(3050\) feet long, how high above ground is the platform where riders begin?
Example 6: Another zipline in West Virginia spans \(1800\) feet across a river gorge. If the platform is \(200\) feet above ground, what is the angle of depression to the base of the zipline?