8.5: Tangent, Sine and Cosine
Learning Objectives
- Use the tangent, sine and cosine ratios in a right triangle.
- Understand these trigonometric ratios in special right triangles.
- Use a scientific calculator to find sine, cosine and tangent.
- Use trigonometric ratios in real-life situations.
Review Queue
- The legs of an isosceles right triangle have length 14. What is the hypotenuse?
- Do the lengths 8, 16, 20 make a right triangle? If not, is the triangle obtuse or acute?
- In a 30-60-90 triangle, what do the 30, 60, and 90 refer to?
- Find the measure of the missing lengths.
Know What? A restaurant needs to build a wheelchair ramp for its customers. The angle of elevation for a ramp is recommended to be \begin{align*}5^\circ\end{align*}. If the vertical distance from the sidewalk to the front door is two feet, what is the horizontal distance that the ramp will take up \begin{align*}(x)\end{align*}? How long will the ramp be \begin{align*}(y)\end{align*}? Round your answers to the nearest hundredth.
What is Trigonometry?
The word trigonometry comes from two words meaning triangle and measure. In this lesson we will define three trigonometric (or trig) functions. Once we have defined these functions, we will be able to solve problems like the Know What? above.
Trigonometry: The study of the relationships between the sides and angles of right triangles.
In trigonometry, sides are named in reference to a particular angle. The hypotenuse of a triangle is always the same, but the terms adjacent and opposite depend on which angle you are referencing. A side adjacent to an angle is the leg of the triangle that helps form the angle. A side opposite to an angle is the leg of the triangle that does not help form the angle. We never reference the right angle when referring to trig ratios.
\begin{align*}a \ \text{is} \ adjacent \ \text{to} \ \angle B. \qquad \ a \ \text{is} \ opposite \ \angle A.\!\\ b \ \text{is} \ adjacent \ \text{to} \ \angle A. \qquad \ b \ \text{is} \ opposite \ \angle B.\!\\ \\ c \ \text{is the} \ hypotenuse.\end{align*}
Sine, Cosine, and Tangent Ratios
The three basic trig ratios are called, sine, cosine and tangent. At this point, we will only take the sine, cosine and tangent of acute angles. However, you will learn that you can use these ratios with obtuse angles as well.
Sine Ratio: For an acute angle \begin{align*}x\end{align*} in a right triangle, the \begin{align*}\sin x\end{align*} is equal to the ratio of the side opposite the angle over the hypotenuse of the triangle.
Using the triangle above, \begin{align*}\sin A = \frac{a}{c}\end{align*} and \begin{align*}\sin B = \frac{b}{c}\end{align*}.
Cosine Ratio: For an acute angle \begin{align*}x\end{align*} in a right triangle, the \begin{align*}\cos x\end{align*} is equal to the ratio of the side adjacent to the angle over the hypotenuse of the triangle.
Using the triangle above, \begin{align*}\cos A = \frac{b}{c}\end{align*} and \begin{align*}\cos B = \frac{a}{c}\end{align*}.
Tangent Ratio: For an acute angle \begin{align*}x\end{align*}, in a right triangle, the \begin{align*}\tan x\end{align*} is equal to the ratio of the side opposite to the angle over the side adjacent to \begin{align*}x\end{align*}.
Using the triangle above, \begin{align*}\tan A = \frac{a}{b}\end{align*} and \begin{align*}\tan B = \frac{b}{a}\end{align*}.
There are a few important things to note about the way we write these ratios. First, keep in mind that the abbreviations \begin{align*}\sin x, \cos x\end{align*}, and \begin{align*}\tan x\end{align*} are all functions. Each ratio can be considered a function of the angle (see Chapter 10). Second, be careful when using the abbreviations that you still pronounce the full name of each function. When we write \begin{align*}\sin x\end{align*} it is still pronounced sine, with a long “\begin{align*}i\end{align*}”. When we write \begin{align*}\cos x\end{align*}, we still say co-sine. And when we write \begin{align*}\tan x\end{align*}, we still say tangent.
An easy way to remember ratios is to use the pneumonic SOH-CAH-TOA.
Example 1: Find the sine, cosine and tangent ratios of \begin{align*}\angle A\end{align*}.
Solution: First, we need to use the Pythagorean Theorem to find the length of the hypotenuse.
\begin{align*}5^2 + 12^2 & = h^2\\ 13 & = h\end{align*}
So, \begin{align*}\sin A = \frac{12}{13}, \cos A =\end{align*}, and \begin{align*}\tan A = \frac{12}{5}\end{align*}.
A few important points:
- Always reduce ratios when you can.
- Use the Pythagorean Theorem to find the missing side (if there is one).
- The tangent ratio can be bigger than 1 (the other two cannot).
- If two right triangles are similar, then their sine, cosine, and tangent ratios will be the same (because they will reduce to the same ratio).
- If there is a radical in the denominator, rationalize the denominator.
Example 2: Find the sine, cosine, and tangent of \begin{align*}\angle B\end{align*}.
Solution: Find the length of the missing side.
\begin{align*}AC^2 + 5^2 & = 15^2\\ AC^2 & = 200\\ AC & = 10 \sqrt{2}\end{align*}
Therefore, \begin{align*}\sin B = \frac{10 \sqrt{2}}{15} = \frac{2 \sqrt{2}}{3}, \cos B = \frac{5}{15} = \frac{1}{3}\end{align*}, and \begin{align*}\tan B = \frac{10 \sqrt{2}}{5} = 2 \sqrt{2}\end{align*}.
Example 3: Find the sine, cosine and tangent of \begin{align*}30^\circ\end{align*}.
Solution: This is a special right triangle, a 30-60-90 triangle. So, if the short leg is 6, then the long leg is \begin{align*}6 \sqrt{3}\end{align*} and the hypotenuse is 12.
\begin{align*}\sin 30^\circ = \frac{6}{12} = \frac{1}{2}, \cos 30^\circ = \frac{6 \sqrt{3}}{12} = \frac{\sqrt{3}}{2}\end{align*}, and \begin{align*}\tan 30^\circ = \frac{6}{6 \sqrt{3}} = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}\end{align*}.
In Example 3, we knew the angle measure of the angle we were taking the sine, cosine and tangent of. This means that the sine, cosine and tangent for an angle are fixed.
Sine, Cosine, and Tangent with a Calculator
We now know that the trigonometric ratios are not dependent on the sides, but the ratios. Therefore, there is one fixed value for every angle, from \begin{align*}0^\circ\end{align*} to \begin{align*}90^\circ\end{align*}. Your scientific (or graphing) calculator knows the values of the sine, cosine and tangent of all of these angles. Depending on your calculator, you should have [SIN], [COS], and [TAN] buttons. Use these to find the sine, cosine, and tangent of any acute angle.
Example 4: Find the indicated trigonometric value, using your calculator.
a) \begin{align*}\sin 78^\circ\end{align*}
b) \begin{align*}\cos 60^\circ\end{align*}
c) \begin{align*}\tan 15^\circ\end{align*}
Solution: Depending on your calculator, you enter the degree first, and then press the correct trig button or the other way around. For TI-83s and TI-84s you press the trig button first, followed by the angle. Also, make sure the mode of your calculator is in DEGREES.
a) \begin{align*}\sin 78^\circ = 0.9781\end{align*}
b) \begin{align*}\cos 60^\circ = 0.5\end{align*}
c) \begin{align*}\tan 15^\circ = 0.2679\end{align*}
Finding the Sides of a Triangle using Trig Ratios
One application of the trigonometric ratios is to use them to find the missing sides of a right triangle. All you need is one angle, other than the right angle, and one side. Let’s go through a couple of examples.
Example 5: Find the value of each variable. Round your answer to the nearest hundredth.
Solution: We are given the hypotenuse, so we would need to use the sine to find \begin{align*}b\end{align*}, because it is opposite \begin{align*}22^\circ\end{align*} and cosine to find \begin{align*}a\end{align*}, because it is adjacent to \begin{align*}22^\circ\end{align*}.
\begin{align*}\sin 22^\circ & = \frac{b}{30} && \quad \ \ \cos 22^\circ = \frac{a}{30}\\ 30 \cdot \sin 22^\circ & = b && 30 \cdot \cos 22^\circ = a\\ b & \approx 11.24 && \qquad \qquad \ a \approx 27.82\end{align*}
Example 6: Find the value of each variable. Round your answer to the nearest hundredth.
Solution: Here, we are given the adjacent leg to \begin{align*}42^\circ\end{align*}. To find \begin{align*}c\end{align*}, we need to use cosine and to find \begin{align*} d \end{align*} we will use tangent.
\begin{align*}\cos 42^\circ & = \frac{9}{c} && \quad \tan 42^\circ = \frac{d}{9}\\ c \cdot \cos 42^\circ & = 9 && 9 \cdot \tan 42^\circ = d\\ c & = \frac{9}{\cos 42^\circ} \approx 12.11 && \qquad \qquad d \approx 8.10\end{align*}
Notice in both of these examples, you should only use the information that you are given. For example, you should not use the found value of \begin{align*}b\end{align*} to find \begin{align*} a \end{align*} (in Example 5) because \begin{align*}b\end{align*} is an approximation. Use exact values to give the most accurate answers. However, in both examples you could have also used the complementary angle to the one given.
Angles of Depression and Elevation
Another practical application of the trigonometric functions is to find the measure of lengths that you cannot measure. Very frequently, angles of depression and elevation are used in these types of problems.
Angle of Depression: The angle measured from the horizon or horizontal line, down.
Angle of Elevation: The angle measure from the horizon or horizontal line, up.
Example 7: An inquisitive math student is standing 25 feet from the base of the Washington Monument. The angle of elevation from her horizontal line of sight is \begin{align*}87.4^\circ\end{align*}. If her “eye height” is 5ft, how tall is the monument?
Solution: We can find the height of the monument by using the tangent ratio and then adding the eye height of the student.
\begin{align*}\tan 87.4^\circ & = \frac{h}{25}\\ h & = 25 \cdot \tan 87.4^\circ = 550.54\end{align*}
Adding 5 ft, the total height of the Washington Monument is 555.54 ft.
According to Wikipedia, the actual height of the monument is 555.427 ft.
Know What? Revisited To find the horizontal length and the actual length of the ramp, we need to use the tangent and sine.
\begin{align*}\tan 5^\circ & = \frac{2}{x} && \sin 5^\circ = \frac{2}{y}\\ x & = \frac{2}{\tan 5^\circ} = 22.86 && \qquad y = \frac{2}{\sin 5^\circ} = 22.95\end{align*}
Review Questions
Use the diagram to fill in the blanks below.
- \begin{align*}\tan D = \frac{?}{?}\end{align*}
- \begin{align*}\sin F = \frac{?}{?}\end{align*}
- \begin{align*}\tan F = \frac{?}{?}\end{align*}
- \begin{align*}\cos F = \frac{?}{?}\end{align*}
- \begin{align*}\sin D = \frac{?}{?}\end{align*}
- \begin{align*}\cos D = \frac{?}{?}\end{align*}
From questions 1-6, we can conclude the following. Fill in the blanks.
- \begin{align*}\cos \underline{\;\;\;\;\;\;}= \sin F\end{align*} and \begin{align*}\sin \underline{\;\;\;\;\;\;} = \cos F\end{align*}
- The sine of an angle is ____________ to the cosine of its ______________.
- \begin{align*}\tan D\end{align*} and \begin{align*}\tan F\end{align*} are ___________ of each other.
Use your calculator to find the value of each trig function below. Round to four decimal places.
- \begin{align*}\sin 24^\circ\end{align*}
- \begin{align*}\cos 45^\circ\end{align*}
- \begin{align*}\tan 88^\circ\end{align*}
- \begin{align*}\sin 43^\circ\end{align*}
Find the sine, cosine and tangent of \begin{align*}\angle A\end{align*}. Reduce all fractions and radicals.
Find the length of the missing sides. Round your answers to the nearest hundredth.
- Kristin is swimming in the ocean and notices a coral reef below her. The angle of depression is \begin{align*}35^\circ\end{align*} and the depth of the ocean, at that point, is 250 feet. How far away is she from the reef?
- The Leaning Tower of Piza currently “leans” at a \begin{align*}4^\circ\end{align*} angle and has a vertical height of 55.86 meters. How tall was the tower when it was originally built?
- The angle of depression from the top of an apartment building to the base of a fountain in a nearby park is \begin{align*}72^\circ\end{align*}. If the building is 78 ft tall, how far away is the fountain?
- William spots a tree directly across the river from where he is standing. He then walks 20 ft upstream and determines that the angle between his previous position and the tree on the other side of the river is \begin{align*}65^\circ\end{align*}. How wide is the river?
- Diego is flying his kite one afternoon and notices that he has let out the entire 120 ft of string. The angle his string makes with the ground is \begin{align*}52^\circ\end{align*}. How high is his kite at this time?
- A tree struck by lightning in a storm breaks and falls over to form a triangle with the ground. The tip of the tree makes a \begin{align*}36^\circ\end{align*} angle with the ground 25 ft from the base of the tree. What was the height of the tree to the nearest foot?
- Upon descent an airplane is 20,000 ft above the ground. The air traffic control tower is 200 ft tall. It is determined that the angle of elevation from the top of the tower to the plane is \begin{align*}15^\circ\end{align*}. To the nearest mile, find the ground distance from the airplane to the tower.
- Critical Thinking Why are the sine and cosine ratios always be less than 1?
Review Queue Answers
- The hypotenuse is \begin{align*}14 \sqrt{2}\end{align*}.
- No, \begin{align*}8^2 + 16^2 < 20^2,\end{align*} the triangle is obtuse.
- \begin{align*}30^\circ, 60^\circ,\end{align*} and \begin{align*}90^\circ\end{align*} refer to the angle measures in the special right triangle.
- \begin{align*}x = 2, y = 2 \sqrt{3}\end{align*}
- \begin{align*}x = 6 \sqrt{3}, y = 18, z = 18 \sqrt{3}, w = 36\end{align*}