Download - Introduction
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IntroductionIn the real world, if you needed to verify the size of a television, you could get some measuring tools and hold them up to the television to determine that the TV was advertised at the correct size. Imagine, however, that you are a fact checker for The Guinness Book of World Records. It is your job to verify that the tallest building in the world is in fact Burj Khalifa, located in Dubai. Could you use measuring tools to determine the size of a building so large? It would be extremely difficult and impractical to do so.
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Introduction, continuedYou can use measuring tools for direct measurements of distance, but you can use trigonometry to find indirect measurements. First, though, you must be able to calculate the three basic trigonometric functions that will be applied to finding those larger distances. Specifically, we are going study and practice calculating the sine, cosine, and tangent functions of right triangles as preparation for measuring indirect distances.
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Key Concepts• The three basic trigonometric ratios are ratios of the
side lengths of a right triangle with respect to one of its acute angles.
• As you learned previously:
• Given the angle
• Given the angle
• Given the angle
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Key Concepts, continued• Notice that the trigonometric ratios contain three
unknowns: the angle measure and two side lengths. • Given an acute angle of a right triangle and the
measure of one of its side lengths, use sine, cosine, or tangent to find another side.
• If you know the value of a trigonometric function, you can use the inverse trigonometric function to find the measure of the angle.
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Key Concepts, continued• The inverse trigonometric functions are arcsine,
arccosine, and arctangent.• Arcsine is the inverse of the sine function, written
• Arccosine is the inverse of the cosine function, written
• Arctangent is the inverse of the tangent function, written
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Key Concepts, continued• There are two different types of notation for these
functions:
• if
• if
• if
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Key Concepts, continued• Note that “–1” is not an exponent; it is simply the
notation for an inverse trigonometric function. Because this notation can lead to confusion, the “arc-” notation is frequently used instead.
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Key Concepts, continued• To calculate inverse trigonometric functions on your
graphing calculator:
On a TI-83/84:Step 1: Press [2ND][SIN]. Step 2: Type in the ratio. Step 3: Press [ENTER].
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Key Concepts, continuedOn a TI-Nspire:
Step 1: In the calculate window from the home screen, press [trig] to bring up the menu of trigonometric functions. Use the keypad to select "sin–1.”
Step 2: Type in the ratio. Step 3: Press [enter].
• Use the inverses of the trigonometric functions to find the acute angle measures given two sides of a right triangle.
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Common Errors/Misconceptions• not using the given angle as the guide in determining
which side is opposite or adjacent • forgetting to ensure the calculator is in degree mode
before completing the operations • using the trigonometric function instead of the inverse
trigonometric function when calculating the acute angle measure
• confusing inverse notation with reciprocal notation
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Guided PracticeExample 2A trucker drives 1,027 feet up a hill that has a constant slope. When the trucker reaches the top of the hill, he has traveled a horizontal distance of 990 feet. At what angle did the trucker drive to reach the top? Round your answer to the nearest degree.
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Guided Practice: Example 2, continued1. Determine which trigonometric function
to use by identifying the given information. Given an angle of w°, the horizontal distance, 990 feet, is adjacent to the angle.
The distance traveled by the trucker is the hypotenuse since it is opposite the right angle of the triangle.
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Guided Practice: Example 2, continued
Cosine is the trigonometric function that uses
adjacent and hypotenuse,
so we will use it to calculate the angle the truck drove to reach the bottom of the road. 13
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Guided Practice: Example 2, continuedSet up an equation using the cosine function and
the given measurements.
Therefore,
Solve for w.
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Guided Practice: Example 2, continuedSolve for w by using the inverse cosine since we
are finding an angle instead of a side length.
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Guided Practice: Example 2, continued2. Use a calculator to calculate the value
of w. On a TI-83/84: First, make sure your calculator is in DEGREE mode:
Step 1: Press [MODE]. Step 2: Arrow down twice to RADIAN. Step 3: Arrow right to DEGREE. Step 4: Press [ENTER]. The word “DEGREE”
should be highlighted inside a black rectangle.
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Guided Practice: Example 2, continuedStep 5: Press [2ND]. Step 6: Press [MODE] to QUIT. Note: You will not have to change to Degree mode again unless you have changed your calculator to Radian mode.
Next, perform the calculation.
Step 1: Press [2ND][COS][990][÷][1027][)].Step 2: Press [ENTER].
w = 15.426, or 15°.
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Guided Practice: Example 2, continuedOn a TI-Nspire: First, make sure the calculator is in Degree mode:
Step 1: Choose 5: Settings & Status, then 2: Settings, and 2: Graphs and Geometry.
Step 2: Move to the Geometry Angle field and choose “Degree”.
Step 3: Press [tab] to “ok” and press [enter].
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Guided Practice: Example 2, continuedThen, if necessary, set the Scratchpad in Degree mode:
Step 1: In the calculate window from the home screen, press [doc].
Step 2: Select 7: Settings and Status, then 2: Settings, and 1: General.
Step 3: Move to the Angle field and choose “Degree”.
Step 4: Press [tab] to “Make Default” and press [enter] twice to apply this as the new default setting.
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Guided Practice: Example 2, continuedNext, perform the calculation.
Step 1: In the calculate window from the home screen, press [trig] to bring up the menu of trigonometric functions. Use the keypad to select "cos–1." Enter 990, then press [÷] and enter 1027.
Step 2: Press [enter].
w = 15.426, or 15°.The trucker drove at an angle of 15° to the top of the hill.
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✔
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Guided Practice: Example 2, continued
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Guided PracticeExample 4Solve the right triangle. Round sides to the nearest thousandth.
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Guided Practice: Example 4, continued1. Find the measures of and .
Solving the right triangle means to find all the missing angle measures and all the missing side lengths. The given angle is 64.5° and 17 is the length of the adjacent side. With this information, we could either use cosine or tangent since both functions’ ratios include the adjacent side of a right triangle. Start by using the tangent function to find .
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Guided Practice: Example 4, continued
Recall that
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Guided Practice: Example 4, continuedOn a TI-83/84:
Step 1: Press [17][TAN][64.5][)]. Step 2: Press [ENTER]. x = 35.641
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Guided Practice: Example 4, continuedOn a TI-Nspire:
Step 1: In the calculate window from the home screen, enter 17, then press [trig] to bring up the menu of trigonometric functions. Use the keypad to select "tan," then enter 64.5.
Step 2: Press [enter]. x = 35.641
The measure of AC = 35.641.
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Guided Practice: Example 4, continuedTo find the measure of , either acute angle may be used as an angle of interest. Since two side lengths are known, the Pythagorean Theorem may be used as well.
Note: It is more precise to use the given values instead of approximated values.
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Guided Practice: Example 4, continued2. Use the cosine function based on the
given information.
Recall that
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Guided Practice: Example 4, continued
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Guided Practice: Example 4, continuedOn a TI-83/84: Check to make sure your calculator is in Degree mode first. Refer to the directions in the previous example.
Step 1: Press [17][÷][COS][64.5][)]. Step 2: Press [ENTER]. y = 39.488
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Guided Practice: Example 4, continuedOn a TI-Nspire: Check to make sure your calculator is in Degree mode first. Refer to the directions in the previous example.
Step 1: In the calculate window from the home screen, enter 17, then press [÷][trig]. Use the keypad to select "cos," and then enter 64.5.
Step 2: Press [enter]. y = 39.488
The measure of AB = 39.488.
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Guided Practice: Example 4, continued3. Use the Pythagorean Theorem to check
your trigonometry calculations.
AC = 35.641 and AB = 39.488.
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The answer checks out.
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Guided Practice: Example 4, continued4. Find the value of ∠A.
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Guided Practice: Example 4, continuedUsing trigonometry, you could choose any of the three functions since you have solved for all three side lengths. In an attempt to be as precise as possible, let’s choose the given side length and one of the approximate side lengths.
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Guided Practice: Example 4, continued5. Use the inverse trigonometric function
since you are solving for an angle measure.
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Guided Practice: Example 4, continuedOn a TI-83/84:
Step 1: Press [2ND][SIN][17][÷][39.488][)]. Step 2: Press [ENTER]. z = 25.500°
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Guided Practice: Example 4, continuedOn a TI-Nspire:
Step 1: In the calculate window from the home screen, press [trig] to bring up the menu of trigonometric functions. Use the keypad to select "sin–1," and then enter 17, press [÷], and enter 39.488.
Step 2: Press [enter].z = 25.500°
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Guided Practice: Example 4, continuedCheck your angle measure by using the Triangle Sum Theorem.
∠A is 25.5°.
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✔The answer checks out.
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Guided Practice: Example 4, continued
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