1.1: Right Triangles
- Page ID
- 102510
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Sine, cosine, tangent, and other ratios of sides of a right triangle.
Sine, Cosine, and Tangent
Trigonometry is the study of the relationships between the sides and angles of right triangles. The legs are called adjacent or opposite depending on which acute angle is being used.
\(\begin{aligned} a \text{ is adjacent to } \angle B \qquad a \text{ is opposite } \angle A \\ b \text{ is adjacent to } \angle A \qquad b \text{ is opposite } \angle B\\ c \text{ is the hypotenuse }\end{aligned}\)
The three basic trigonometric ratios are called sine, cosine and tangent. For right triangle △ABC, we have:
\(\begin{aligned}
&\text { sine Ratio: } \dfrac{\text {opposite leg}}{\text {hypotenuse}} \qquad \sin A=\dfrac{a}{c} \text { or } \sin B=\dfrac{b}{c}\\
&\text { cosine Ratio: } \dfrac{\text {adjacent leg}}{\text {hypotenuse}} \qquad \cos A=\dfrac{b}{c} \text { or } \cos B=\dfrac{a}{c}\\
&\text { Tangent Ratio: } \dfrac{\text {opposite leg}}{\text {adjacent leg}} \qquad \tan A=\dfrac{a}{b} \text { or } \tan B=\dfrac{b}{a}
\end{aligned}\)
An easy way to remember ratios is to use SOH-CAH-TOA.
A few important points:
- Always reduce ratios (fractions) when you can.
- Use the Pythagorean Theorem to find the missing side (if there is one).
- If there is a radical in the denominator, rationalize the denominator.
What if you were given a right triangle and told that its sides measure 3, 4, and 5 inches? How could you find the sine, cosine, and tangent of one of the triangle's non-right angles?
Find the sine, cosine and tangent ratios of \(\angle A\).
Solution
First, we need to use the Pythagorean Theorem to find the length of the hypotenuse.
\(\begin{aligned}
5^{2}+12^{2} &=c^{2} \\
13 &=c \\
\sin A &=\frac{l e g \text { opposite } \angle A}{\text {hypotenuse}}=\frac{12}{13} & \cos A=\frac{\text {leg adjacent to } \angle A}{\text {hypotenuse}}=\frac{5}{13}, \\
\tan A &=\frac{\text {leg opposite } \angle A}{\text {leg adjacent to } \angle A}=\frac{12}{5}
\end{aligned}\)
Find the sine, cosine, and tangent of \(\angle B\).
Find the length of the missing side.
Solution
\(\begin{aligned}
A C^{2}+5^{2} &=15^{2} \\
A C^{2} &=200 \\
A C &=10 \sqrt{2} \\
\sin B &=\frac{10 \sqrt{2}}{15}=\frac{2 \sqrt{2}}{3} \quad \cos B=\frac{5}{15}=\frac{1}{3} \quad \tan B=\frac{10 \sqrt{2}}{5}=2 \sqrt{2}
\end{aligned}\)
Find the sine, cosine and tangent of \(30^{\circ}\).
Solution
This is a 30-60-90 triangle. The short leg is 6, \(y=6\sqrt{3}\) and \(x=12\).
\(\sin 30^{\circ}=\dfrac{6}{12}=\dfrac{1}{2} \qquad \cos 30^{\circ}=\dfrac{6\sqrt{3}}{12}=\dfrac{\sqrt{3}}{2} \qquad \tan 30^{\circ}=\dfrac{6}{6\sqrt{3}}=\dfrac{1}{\sqrt{3}}\cdot \dfrac{\sqrt{3}}{\sqrt{3}}=\dfrac{\sqrt{3}}{3}\)
Answer the questions about the following image. Reduce all fractions.
What is sin A, cos A, and tan A?
Solution
\(\begin{array}{l}
\sin A=\frac{16}{20}=\frac{4}{5} \\
\cos A=\frac{12}{20}=\frac{3}{5} \\
\tan A=\frac{16}{12}=\frac{4}{3}
\end{array}\)
Resources
Vocabulary
| Term | Definition |
|---|---|
| Acute Angle | An acute angle is an angle with a measure of less than 90 degrees. |
| Adjacent Angles | Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'next-to'. |
| Hypotenuse | The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle. |
| Legs of a Right Triangle | The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle. |
| opposite | The opposite of a number \(x\) is \(−x\). A number and its opposite always sum to zero. |
| Pythagorean Theorem | The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by \(a^2+b^2=c^2\), where a and b are legs of the triangle and c is the hypotenuse of the triangle. |
| Radical | The \(\sqrt{}\), or square root, sign. |
| sine | The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse. |
| Trigonometric Ratios | Ratios that help us to understand the relationships between sides and angles of right triangles. |
Additional Resources
Video: Introduction to Trigonometric Functions Using Triangles
Activities: Sine, Cosine, Tangent Discussion Questions
Study Aids: Trigonometric Ratios Study Guide
Practice: Right Triangle Trigonometry
Real World: Sine Cosine Tangent
The Pythagorean Therorem
Pythagoras was a Greek mathematician and philosopher, born on the island of Samos (ca. 582 BC). He founded a number of schools, one in particular in a town in southern Italy called Crotone, whose members eventually became known as the Pythagoreans. The inner circle at the school, the Mathematikoi, lived at the school, rid themselves of all personal possessions, were vegetarians, and observed a strict vow of silence. They studied mathematics, philosophy, and music, and held the belief that numbers constitute the true nature of things, giving numbers a mystical or even spiritual quality.
Today, nothing is known of Pythagoras’s writings, perhaps due to the secrecy and silence of the Pythagorean society. However, one of the most famous theorems in all of mathematics does bear his name, the Pythagorean Theorem.
Prior to revealing the contents of the Pythagorean Theorem, we pause to provide the definition of a right triangle and its constituent parts.
A triangle with one right angle (90◦) is called a right triangle. In the figure below, the right angle is marked with a little square.
The side of the triangle that is directly opposite the right angle is called the hypotenuse. The sides of the triangle that include the right angle are called the legs of the right triangle.
Now we can state one of the most ancient theorems of mathematics, the Pythagorean Theorem.
The relationship involving the legs and hypotenuse of the right triangle, given by
\[a^2 + b^2 = c^2,\nonumber \]
is called the Pythagorean Theorem.
Here are two important observations.
Two important facts regarding the hypotenuse of the right triangle are:
- The hypotenuse is the longest side of the triangle and lies directly opposite the right angle.
- In the Pythagorean equation \(a^2 + b^2 = c^2\), the hypotenuse lies by itself on one side of the equation.
*The Pythagorean Theorem can only be applied to right triangles.*
Let’s look at a simple application of the Pythagorean Theorem.
The legs of a right triangle measure 3 and 4 meters, respectively. Find the length of the hypotenuse.
Solution
Let’s follow the Requirements for Word Problem Solutions.
1. Set up a Variable Dictionary. Let c represent the length of the hypotenuse, as pictured in the following sketch.
2. Set up an Equation. The Pythagorean Theorem says that
\[a^2 + b^2 = c^2.\nonumber \]
In this example, the legs are known. Substitute 4 for a and 3 for b (3 for a and 4 for b works equally well) into the Pythagorean equation.
\[4^2 + 3^2 = c^2\nonumber \]
3. Solve the Equation.
\[ \begin{aligned} 4^2 + 3^2 = c^2 ~ & \textcolor{red}{ \text{ The Pythagorean equation.}} \\ 16 + 9 = c^2 ~ & \textcolor{red}{ \text{ Exponents first: } 4^2 = 16 \text{ and } 3^2 = 9.} \\ 25 = c^2 ~ & \textcolor{red}{ \text{ Add: } 16+9=25.} \\ 5 = c~ & \textcolor{red}{ \text{ Take the nonnegative square root.}} \end{aligned}\nonumber \]
Technically, there are two answers to c2 = 25, i.e., c = −5 or c = 5. However, c represents the hypotenuse of the right triangle and must be nonnegative. Hence, we must choose c = 5.
4. Answer the Question. The hypotenuse has length 5 meters.
5. Look Back. Do the numbers satisfy the Pythagorean Theorem? The sum of the squares of the legs should equal the square of the hypotenuse. Let’s check.
\[\begin{aligned} 4^2 + 3^2 = 5^2 \\ 16 + 9 = 25 \\ 25 = 25 \end{aligned}\nonumber \]
All is well!
Sometimes an approximate answer is desired, particularly in applications.
Ginny want to create a vegetable garden in the corner of her yard in the shape of a right triangle. She cuts two boards of length 8 feet which will form the legs of her garden. Find the length of board she should cut to form the hypotenuse of her garden, correct to the nearest tenth of a foot.
Solution
1. Set Up an Equation. The hypotenuse is isolated on one side of the Pythagorean equation.
\[x^2 = 8^2 + 8^2\nonumber \]
2. Solve the Equation.
\[ \begin{aligned} x^2 = 8^2 + 8^2 ~ & \textcolor{red}{ \text{ The Pythagorean equation.}} \\ x^2 = 64 + 64 ~ & \textcolor{red}{ \text{ Exponents first: } 8^2 = 64 \text{ and } 8^2 = 64.} \\ x^2 = 128 ~ & \textcolor{red}{ \text{ Add: } 64 + 64 = 128.} \\ x = \sqrt{128} ~ & \textcolor{red}{ \text{ Take the nonnegative square root.}} \end{aligned}\nonumber \]
3. Answer the Question. The exact length of the hypotenuse is \(\sqrt{128}\) feet, but we’re asked to find the hypotenuse to the nearest tenth of a foot. Using a calculator, we find an approximation for \(\sqrt{128}\).
\[\sqrt{128} \approx 11.313708499\nonumber \]
To round to the nearest tenth, first identify the rounding and test digits.
The test digit is less than five. So we leave the rounding digit alone and truncate. Therefore, correct to the nearest tenth of a foot, the length of the hypotenuse is approximately 11.3 feet.
4. Look Back. The sum of the squares of the legs is
\[ \begin{aligned} 8^2 + 8^2 = 64 + 64 \\ = 128. \end{aligned}\nonumber \]
The square of the hypotenuse is
\[(11.3)^2 = 127.69\nonumber \]
These are almost the same, the discrepancy due to the fact that we rounded to find an approximation for the hypotenuse.
Exercises
In Exercises 1-16, your solutions should include a well-labeled sketch.
1. The length of one leg of a right triangle is 15 meters, and the length of the hypotenuse is 25 meters. Find the exact length of the other leg.
2. The length of one leg of a right triangle is 7 meters, and the length of the hypotenuse is 25 meters. Find the exact length of the other leg.
3. The lengths of two legs of a right triangle are 12 meters and 16 meters. Find the exact length of the hypotenuse.
4. The lengths of two legs of a right triangle are 9 meters and 12 meters. Find the exact length of the hypotenuse.
5. The length of one leg of a right triangle is 13 meters, and the length of the hypotenuse is 22 meters. Find the exact length of the other leg.
6. The length of one leg of a right triangle is 6 meters, and the length of the hypotenuse is 15 meters. Find the exact length of the other leg.
7. The lengths of two legs of a right triangle are 2 meters and 21 meters. Find the exact length of the hypotenuse.
8. The lengths of two legs of a right triangle are 7 meters and 8 meters. Find the exact length of the hypotenuse.
9. The length of one leg of a right triangle is 12 meters, and the length of the hypotenuse is 19 meters. Find the exact length of the other leg.
10. The length of one leg of a right triangle is 5 meters, and the length of the hypotenuse is 10 meters. Find the exact length of the other leg.
11. The lengths of two legs of a right triangle are 6 meters and 8 meters. Find the exact length of the hypotenuse.
12. The lengths of two legs of a right triangle are 5 meters and 12 meters. Find the exact length of the hypotenuse.
13. The length of one leg of a right triangle is 6 meters, and the length of the hypotenuse is 10 meters. Find the exact length of the other leg.
14. The length of one leg of a right triangle is 9 meters, and the length of the hypotenuse is 15 meters. Find the exact length of the other leg.
15. The lengths of two legs of a right triangle are 6 meters and 22 meters. Find the exact length of the hypotenuse.
16. The lengths of two legs of a right triangle are 9 meters and 19 meters. Find the exact length of the hypotenuse.
In Exercises 17-24, your solutions should include a well-labeled sketch.
17. The lengths of two legs of a right triangle are 3 meters and 18 meters. Find the length of the hypotenuse. Round your answer to the nearest hundredth.
18. The lengths of two legs of a right triangle are 10 feet and 16 feet. Find the length of the hypotenuse. Round your answer to the nearest tenth.
19. The length of one leg of a right triangle is 2 meters, and the length of the hypotenuse is 17 meters. Find the length of the other leg. Round your answer to the nearest tenth.
20. The length of one leg of a right triangle is 4 meters, and the length of the hypotenuse is 12 meters. Find the length of the other leg. Round your answer to the nearest hundredth.
21. The lengths of two legs of a right triangle are 15 feet and 18 feet. Find the length of the hypotenuse. Round your answer to the nearest hundredth.
22. The lengths of two legs of a right triangle are 6 feet and 13 feet. Find the length of the hypotenuse. Round your answer to the nearest tenth.
23. The length of one leg of a right triangle is 4 meters, and the length of the hypotenuse is 8 meters. Find the length of the other leg. Round your answer to the nearest hundredth.
24. The length of one leg of a right triangle is 3 meters, and the length of the hypotenuse is 15 meters. Find the length of the other leg. Round your answer to the nearest tenth.
25. Greta and Fritz are planting a 13-meter by 18-meter rectangular garden, and are laying it out using string. They would like to know the length of a diagonal to make sure that right angles are formed. Find the length of a diagonal. Round your answer to the nearest hundredth. Your solution should include a well-labeled sketch.
26. Markos and Angelina are planting an 11- meter by 19-meter rectangular garden, and are laying it out using string. They would like to know the length of a diagonal to make sure that right angles are formed. Find the length of a diagonal. Round your answer to the nearest tenth. Your solution should include a well-labeled sketch.
27. The base of a 24-meter long guy wire is located 10 meters from the base of the telephone pole that it is anchoring. How high up the pole does the guy wire reach? Round your answer to the nearest hundredth. Your solution should include a well-labeled sketch.
28. The base of a 30-foot long guy wire is located 9 feet from the base of the telephone pole that it is anchoring. How high up the pole does the guy wire reach? Round your answer to the nearest hundredth. Your solution should include a well-labeled sketch.
29. Hiking Trail. A hiking trail runs due south for 8 kilometers, then turns west for about 15 kilometers, and then heads northeast on a direct path to the starting point. How long is the entire trail?
30. Animal Trail. An animal trail runs due east from a watering hole for 12 kilometers, then goes north for 5 kilometers. Then the trail turns southwest on a direct path back to the watering hole. How long is the entire trail?
31. Upper Window. A 10-foot ladder leans against the wall of a house. How close to the wall must the bottom of the ladder be in order to reach a window 8 feet above the ground?
32. How high? A 10-foot ladder leans against the wall of a house. How high will the ladder be if the bottom of the ladder is 4 feet from the wall? Round your answer to the nearest tenth.
Answers
1. 20 meters
3. 20 meters
5. √315 meters
7. \(\sqrt{445}\) meters
9. \(\sqrt{217}\) meters
11. 10 meters
13. 8 meters
15. \(\sqrt{520}\) meters
17. 18.25 meters
19. 16.9 meters
21. 23.43 feet
23. 6.93 meters
25. 22.20 meters
27. 21.82 meters
29. 40 kilometers
31. 6 ft.