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8.5: Tangent, Sine and Cosine

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

  • Use the tangent, sine and cosine ratios.
  • Use a scientific calculator to find sine, cosine and tangent.
  • Use trigonometric ratios in real-life situations.

Review Queue

  1. The legs of an isosceles right triangle have length 14. What is the hypotenuse?
  2. Do the lengths 8, 16, 20 make a right triangle? If not, is the triangle obtuse or acute?
  3. In a 30-60-90 triangle, what do the 30, 60, and 90 refer to?

Know What? A restaurant is building a wheelchair ramp. The angle of elevation for the ramp is \begin{align*}5^\circ\end{align*}5. If the vertical distance from the sidewalk to the front door is 4 feet, how long will the ramp be \begin{align*}(x)\end{align*}(x)? Round your answers to the nearest hundredth.

What is Trigonometry?

In this lesson we will define three trigonometric (or trig) ratios. Once we have defined these ratios, 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.

The legs are called adjacent or opposite depending on which acute angle is being used.

\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*}a is adjacent to B a is opposite Ab is adjacent to A b is opposite Bc is the hypotenuse

Sine, Cosine, and Tangent Ratios

The three basic trig ratios are called, sine, cosine and tangent. For now, we will only take the sine, cosine and tangent of acute angles. However, you can use these ratios with obtuse angles as well.

For right triangle \begin{align*}\triangle ABC\end{align*}ABC, we have:

Sine Ratio: \begin{align*}\frac{opposite \ leg }{hypotenuse} \ \sin A = \frac{a}{c}\end{align*}opposite leghypotenuse sinA=ac or \begin{align*}\sin B = \frac{b}{c}\end{align*}sinB=bc

Cosine Ratio: \begin{align*}\frac{adjacent \ leg}{hypotenuse} \ \cos A = \frac{b}{c}\end{align*}adjacent leghypotenuse cosA=bc or \begin{align*}\cos B = \frac{a}{c}\end{align*}cosB=ac

Tangent Ratio: \begin{align*}\frac{opposite \ leg}{adjacent \ leg} \ \tan A = \frac{a}{b}\end{align*}opposite legadjacent leg tanA=ab or \begin{align*}\tan B = \frac{b}{a}\end{align*}tanB=ba

An easy way to remember ratios is to use SOH-CAH-TOA.

Example 1: Find the sine, cosine and tangent ratios of \begin{align*}\angle A\end{align*}A.

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\\ \sin A &= \frac{leg \ opposite \ \angle A}{hypotenuse} = \frac{12}{13} && \cos A = \frac{leg \ adjacent \ to \ \angle A}{hypotenuse}= \frac{5}{13},\\ \tan A &= \frac{leg \ opposite \ \angle A}{leg \ adjacent \ to \ \angle A}= \frac{12}{5}\end{align*}52+12213sinAtanA=h2=h=leg opposite Ahypotenuse=1213=leg opposite Aleg adjacent to A=125cosA=leg adjacent to Ahypotenuse=513,

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.

Example 2: Find the sine, cosine, and tangent of \begin{align*}\angle B\end{align*}B.

Solution: Find the length of the missing side.

\begin{align*}AC^2 + 5^2 &= 15^2\\ AC^2 &= 200\\ AC &= 10 \sqrt{2}\\ \sin B &= \frac{10 \sqrt{2}}{15} = \frac{2 \sqrt{2}}{3} && \cos B = \frac{5}{15}=\frac{1}{3} && \tan B = \frac{10 \sqrt{2}}{5} = 2 \sqrt{2}\end{align*}AC2+52AC2ACsinB=152=200=102=10215=223cosB=515=13tanB=1025=22

Example 3: Find the sine, cosine and tangent of \begin{align*}30^\circ\end{align*}30.

Solution: This is a 30-60-90 triangle. The short leg is 6, \begin{align*}y = 6 \sqrt{3}\end{align*}y=63 and \begin{align*}x=12\end{align*}x=12.

\begin{align*}\sin 30^\circ = \frac{6}{12} = \frac{1}{2} && \cos 30^\circ = \frac{6 \sqrt{3}}{12} = \frac{3}{2} && \tan 30^\circ = \frac{6}{6 \sqrt{3}} = \frac{1}{\sqrt{3}} \cdot \frac{3}{\sqrt{3}} = \frac{\sqrt{3}}{3}\end{align*}sin30=612=12cos30=6312=32tan30=663=1333=33

Sine, Cosine, and Tangent with a Calculator

From Example 3, we can conclude that there is a fixed sine, cosine, and tangent value for every angle, from \begin{align*}0^\circ\end{align*}0 to \begin{align*}90^\circ\end{align*}90. Your scientific (or graphing) calculator knows all the trigonometric values for any angle. Your calculator, should have [SIN], [COS], and [TAN] buttons.

Example 4: Find the trigonometric value, using your calculator. Round to 4 decimal places.

a) \begin{align*}\sin 78^\circ\end{align*}sin78

b) \begin{align*}\cos 60^\circ\end{align*}cos60

c) \begin{align*}\tan 15^\circ\end{align*}tan15

Solution: Depending on your calculator, you enter the degree and then press the trig button or the other way around. Also, make sure the mode of your calculator is in DEGREES.

a) \begin{align*}\sin 78^\circ = 0.9781\end{align*}sin78=0.9781

b) \begin{align*}\cos 60^\circ = 0.5\end{align*}cos60=0.5

c) \begin{align*}\tan 15^\circ = 0.2679\end{align*}tan15=0.2679

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.

Example 5: Find the value of each variable. Round your answer to the nearest tenth.

Solution: We are given the hypotenuse. Use sine to find \begin{align*}b\end{align*}b, and cosine to find \begin{align*}a\end{align*}a.

\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 11.2 && \qquad \quad \ \ \ a \approx 27.8\end{align*}sin2230sin22b=b30=b11.2  cos22=a3030cos22=a   a27.8

Example 6: Find the value of each variable. Round your answer to the nearest tenth.

Solution: We are given the adjacent leg to \begin{align*}42^\circ\end{align*}42. To find \begin{align*}c\end{align*}c, use cosine and tangent to find \begin{align*}d\end{align*}d.

\begin{align*}\cos 42^\circ &= \frac{adjacent}{hypotenuse} = \frac{9}{c} && \quad \tan 42^\circ = \frac{opposite}{adjacent} = \frac{d}{9}\\ c \cdot \cos 42^\circ &= 9 && 9 \cdot \tan 42^\circ = d\\ c &= \frac{9}{\cos 42^\circ} \approx 12.1 && \qquad \quad \ \ d \approx 8.1\end{align*}cos42ccos42c=adjacenthypotenuse=9c=9=9cos4212.1tan42=oppositeadjacent=d99tan42=d  d8.1

Anytime you use trigonometric ratios, only use the information that you are given. This will give the most accurate answers.

Angles of Depression and Elevation

Another application of the trigonometric ratios is to find 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: A 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 5 ft, how tall is the monument?

Solution: We can find the height of the monument by using the tangent ratio.

\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.

Know What? Revisited To find the length of the ramp, we need to use sine.

\begin{align*}\sin 5^\circ &= \frac{4}{x}\\ y &= \frac{2}{\sin 5^\circ} = 22.95\end{align*}

Review Questions

  • Questions 1-8 use the definitions of sine, cosine and tangent.
  • Questions 9-16 are similar to Example 4.
  • Questions 17-22 are similar to Examples 1-3.
  • Questions 23-28 are similar to Examples 5 and 6.
  • Questions 29 and 30 are similar to Example 7.

Use the diagram to fill in the blanks below.

  1. \begin{align*}\tan D = \frac{?}{?}\end{align*}
  2. \begin{align*}\sin F = \frac{?}{?}\end{align*}
  3. \begin{align*}\tan F = \frac{?}{?}\end{align*}
  4. \begin{align*}\cos F = \frac{?}{?}\end{align*}
  5. \begin{align*}\sin D = \frac{?}{?}\end{align*}
  6. \begin{align*}\cos D = \frac{?}{?}\end{align*}

From questions 1-6, we can conclude the following. Fill in the blanks.

  1. \begin{align*}\cos \underline{\;\;\;\;\;\;\;} = \sin F\end{align*} and \begin{align*}\sin \underline{\;\;\;\;\;\;\;} = \cos F\end{align*}.
  2. \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.

  1. \begin{align*}\sin 24^\circ\end{align*}
  2. \begin{align*}\cos 45^\circ\end{align*}
  3. \begin{align*}\tan 88^\circ\end{align*}
  4. \begin{align*}\sin 43^\circ\end{align*}
  5. \begin{align*}\tan 12^\circ\end{align*}
  6. \begin{align*}\cos 79^\circ\end{align*}
  7. \begin{align*}\sin 82^\circ\end{align*}
  8. \begin{align*}\tan 45^\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 tenth.

  1. 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?
  2. 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?

Review Queue Answers

  1. The hypotenuse is \begin{align*}14 \sqrt{2}\end{align*}.
  2. No, \begin{align*}8^2 + 16^2 < 20^2\end{align*}, the triangle is obtuse.
  3. \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.

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