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1.8: Right Triangle Trigonometry

Created by: CK-12

The Pythagorean Theorem

Pacing: This lesson should take one class period

Goal: This lesson introduces the Pythagorean Theorem. It is arguably one of the most important theorems in mathematics, allowing for a multitude of uses, including the Law of Cosines.

Visualization! Here is a second proof of Pythagorean’s Theorem that students can do in class.

  1. Reproduce the following diagram for each student.
  2. Students cut the red square and the blue square from the triangle.
  3. Cut along the lines within the blue square. Students should have four pieces.
  4. Students will fit these four puzzle pieces onto the yellow triangle, proving that the combined area of the two smaller squares equal the area of the largest square. Hence, a^2 + b^2 = c^2

Illustration_to_Euclid's_proof_of_the_Pythagorean_theorem.svg

Additional Examples:

  1. A rectangular park measures 500\;\mathrm{m} by 650\;\mathrm{m}. How much shorter is the path diagonally than walking around the outside edge?
  2. Television sets are described according to its diagonal length. A 42” TV means the diagonal of the screen is 42” long. Suppose the TV below is 36” tall with a 47” diagonal. How wide is the TV?

Converse of the Pythagorean Theorem

Pacing: This lesson should take one class period

Goal: This lesson applies the converse of Pythagorean’s Theorem to determine whether triangles are right, acute, or obtuse.

Additional Examples:

  1. Can the following lengths form a right triangle? Explain your answer. 10,15,225. No, 10^2 + 15^2 = 325. However, this will be much less than 225^2.
  2. Find an integer such that the three lengths represent an acute triangle: 9, 12, ____. Sample: 16, 17, 18, 22 \ldots
  3. Find an integer such that the three lengths represent an obtuse triangle: 8, 19, ____. Sample: 20, 10, 14,\ldots

Using Similar Right Triangles

Pacing: This lesson may take two class periods, due to the difficulty of the material

Goal: The concept of geometric mean is used in Advanced Algebra to determine the mean of a widespread data set. In geometry, the geometric mean is illustrated using a right triangle and its altitude.

Look Out! The concept of geometric mean is easy to comprehend, but difficult for student to apply. Spend time in class reviewing this lesson and using additional examples.

An alternative to the abstract formula for geometric mean is, “The altitude of the hypotenuse equals the geometric mean between the segments of the hypotenuse.”

Additional Examples:

  1. Consider the diagram below. Suppose h = 12 and x = 8. Find y.\ y=18
  2. Consider the diagram below. Suppose a = ?, x = 9 and y = 11. Find the value of a. a=6 \sqrt{5}
  3. Consider the diagram below. Suppose x=5 and y = 15. Find the value of the altitude, h. h = 5 \sqrt{3}

Special Right Triangles

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to encourage the use of shortcuts to find values of special right triangles. These triangles are extremely useful when relating the trigonometric functions to the exact values found within the unit circle.

If students have trouble remembering these special shortcuts, encourage them to use Pythagorean’s Theorem and simplify the answer. The resulting answer will equal the shortcut.

In-class Activity! Separate your class into six to ten groups. Write six to ten numbers on the board, one for each group. Instruct the groups to draw an isosceles right triangle with legs of the given length. Have the groups solve for the hypotenuse and share with the remaining groups.

What is the Connection? In any right isosceles triangle, if a leg has value of x, then by Pythagorean’s Theorem, x^2 + x^2 = h^2. Adding like terms, you get 2x^2 = h^2. To solve for the value of the hypotenuse, you must square root both sides, leaving the equation x \sqrt{2} = h. Therefore, the length of the hypotenuse in ANY right isosceles triangle is equal to \mathrm{leg} \sqrt{2}

In any 30-60-90 triangle, the relationship between the segments is as follows: Let the smallest leg have the value of d. The hypotenuse will always have length 2d and the other leg will always have length d\sqrt{3}. This again can be proved using Pythagorean’s Theorem.  

Tangent Ratios

Pacing: This lesson should take one class period

Goal: This lesson introduces the first trigonometric function, the tangent ratio. The tangent seems to be the most natural for students to understand, as opposed to the sine or cosine functions. Tangent ratios occur in many careers, from construction to machine operators.

Check Your Tech! If you are using calculators for the tangent (TAN), cosine (COS), and sine (SIN) functions, be sure to do a “Mode Check.” Have each student check to ensure their calculator is set to degrees (DEG) instead of radians (RAD). Having a calculator in radians will provide incorrect answers and students at this level do not know what radians are to correct their answers.

Vocabulary Connection! Students must understand adjacent and opposite to be successful with trigonometric ratios. They have already had experience with adjacent in previous chapters. Begin by reviewing such vocabulary as adjacent angles and adjacent sides in regards to parallel lines and transversals.

Beginning Activity! Once the class has reviewed the term adjacent, offer students these triangles. Ask students to write the terms adjacent and opposite above the appropriate legs and label the hypotenuse. Always stress that the information stems from the given angle (not the 90\;\mathrm{degree})!

Extension! When discussing tangent values of common angles, such as 30^\circ, 45^\circ, and 60^\circ, review how to rationalize the denominator with your students. This concept should have been presented in Algebra 1. While not as common with the increased use of technology, most standardized test questions will present answers in completely simplified form. For example, the tangent (30^\circ) = \frac{1}{\sqrt{3}}. The objective of rationalizing a denominator is to clear it of decimals, radicals, and complex values. To do so, multiply the fraction by a value of 1, in this case, \frac{\sqrt{3}}{\sqrt{3}}. The new expression becomes \frac{\sqrt{3}}{\sqrt{3}}, a simplified version of the tangent of 30\;\mathrm{degrees}. Additional Examples:

  1. Pradnya’s kite is 50’ away from her in the sky, forming a 27^\circ angle with the ground. Pradnya is 4' \ 6" tall. How high is the kite from the ground? Approximately 30’
  2. The roof pitch is always described in terms of rise/run. Suppose the roof makes a 65^\circ with the horizontal truss and forms the triangle below. How tall is the peak of the roof? 37.53’

Arts and Crafts Time! Using paper, have each student create an astrolabe and use it to determine the height of a tall object, such as a skyscraper or tree.

  1. Begin by folding an 8.5” \times 11” sheet of notebook paper into a square and remove the excess.
  2. Bisect on angle of the square – the segment represents a 45-\mathrm{degree} angle.
  3. Bisect each 45-\mathrm{degree} angle. There should be three creases – two 22.5\;\mathrm{degree} angles and one 45-\mathrm{degree} angle.
  4. Punch a hole in the opposite corner. Tie string through this hole and attach a pencil at the other end.
  5. Go outside and line your astrolabe to the top of something, say a tree. Pretend you are hunting and plan to attack something with your astrolabe.
  6. Gravity will show you the degree of your sight.
  7. Use trigonometric functions to determine its height.

For more information, go to Berkley’s website: http://cse.ssl.berkeley.edu/AtHomeAstronomy/activity_07.html

Sine and Cosine Ratios

Pacing: This lesson should take one to two class periods

Goal: The objective of this lesson is to complete the introduction of trigonometric functions by presenting the sine and cosine ratios.

Welcome to Camp SOH-CAH-TOA! To make trigonometry fun, invite students to Camp SOH-CAH-TOA. Wear a camp counselor outfit, arrange your students in a circle around a makeshift campfire, and begin with an old-fashioned tent (one you can use to point out right triangles).

Have students sketch your tent, splitting it into two right triangles at the altitude (good use of vocabulary!). State that the angle the tent makes to the ground is 55^\circ (something you cannot use special triangles for). Ask students to label each triangle with the appropriate terms: adjacent leg, opposite leg, and hypotenuse. The question is, “How long is the outside edge of the tent?” Question why the tangent ratio cannot be used (the question you want to answer is not the opposite nor adjacent leg). Ask for additional ways to solve the problem. Present the sine and cosine ratios.

Additional Examples/Extensions:

  1. Given the triangle below, find \mathrm{sine}(A) and \mathrm{cos}(B). What is special about these two answers?
  2. Evaluate (\cos(65))^2 + (\mathrm{sine}(65))^2. List as much as you can about this expression and the answer you received. Generalize this question. (\cos(65))^2 + (\mathrm{sine}(65))^2 = 1, the degrees of the sine and cosines are the same value, so the cosine of an angle square plus the sine of an angle squared should equal 1.
  3. Suppose a fireman’s ladder is 39’ long is placed against the side of a building at a 62\;\mathrm{degree} angle. How high will the ladder reach? 18.31’

Inverse Trigonometric Functions

Pacing: This lesson should take one to two class periods

Goal: In the previous two lessons, students used the special trigonometric values to determine approximate angle measurements. This lesson enables students to “cancel” a trigonometric function by applying its inverse to accurately find an angle measurement.

Using Previous Knowledge! Begin by listing several mathematical operations on the board in one column. In a second column, head it with “Inverse.” Be sure students understand what an inverse means (an inverse cancels an operation, leaving the original value undisturbed).

For example,

OPERATION INVERSE
Addition
Squaring
Division
Subtraction
Tangent
Sine

The first four are typically easy for students (Subtraction, square root, multiplication, and addition). You may have to lead students a little more on the last two (inverse tangent and inverse sine). Students may say, “Un-tangent it.” Use the correct terminology here, but also use their wording, if at all possible. Students will be able to cancel the trigonometric function using the inverse of that function, even though they may use incorrect terminology.

Outside at Camp SOH-CAH-TOA! Find the angle of inclination of the sun! Students love this activity, as it gets them outside and applying mathematics in real life. Explain how the Earth progresses around the sun, giving seasons. Also explain how the Earth’s tilt lends to the number of hours of daylight. The combination of these principles describes the angle of elevation (inclination) of the sun in the sky. Create groups of three or four. One student is the statue, one student is the surveyor, and the third is the secretary. Once outside, the statue stands on a flat plane while the surveyor measures the statue’s height and shadow length in the same units and relays this information to the secretary. The secretary draws a right-triangle sketch of the statue and shadow. The goal is to find the angle of elevation (the angle made between the horizon and sun).

Acute and Obtuse Triangles

Pacing: This lesson could take two to four class periods

Goal: The purpose of this lesson is to extend trigonometric ratios to non-right triangles. This is done using to new laws: the Law of Sines and the Law of Cosines. The Law of Sines is much easier to present to students than the Law of Cosines. The Law of Sines uses proportions, while the Law of Cosines uses a general form of Pythagorean’s Theorem.

Shortcut! Students confuse themselves regarding which law to use. A shortcut to use is as follows: If the triangle has more side information than angle information, use the Law of Cosines. If the triangle has more angle information than side information, use the Law of Sines. And of course, if you cannot solve it with the law you have chosen, “Choose the Other One!”

Relate to Triangle Similarity! Have students recall the five basic types of triangle similarity: SSS, SAS, AAS, AAA, ASA. Reading the given information from left to right, if the information ends in an “angle,” use the Law of Sines. If the information ends in a “side,” use the Law of Cosines.

Additional Examples:

1. A fire is spotted in Yellowstone National Park by two forest ranger stations. Fire Station A is 15\;\mathrm{km} from Fire Station B. The angle at which the fire is spotted by Fire Station B is 75 and the angle at which the fire is spotted by Fire Station A is 70\;\mathrm{degrees.} Which fire station should report to the fire? AB = 25.26 \;\mathrm{km} and BC = 24.57. Therefore, Fire Station A should report to the fire.

2. Suppose \triangle ABC has the following values: m \angle C = 40, a = 8, b = 9. Find c. c = 5.89

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CK.MAT.ENG.TE.1.Geometry.1.8

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