Trigonometry Basics Right Triangle Trigonometry Greek Letter Pronounced

Trigonometry Basics Right Triangle Trigonometry

Greek Letter Pronounced “theta” Represents an unknown angle

Pronounced “alpha” Represents an unknown angle

Pronounced “beta” Represents an unknown angle

Pronounced “sigma” Represents an unknown angle

The 3 basic. Trigonometric Functions SINE COSINE TANGENT

The Trigonometric Functions (abbreviations) SINE COSINE TANGENT

Finding the ratios The simplest form of question is finding the decimal value of the ratio of a given angle using your calculator. They will look similar to this. Find using calculator: 1) sin 30 = 2) cos 23 = 3) tan 78 = 4) tan 27 = 5) sin 68 =

Sine Function For example to evaluate sin 40°… Make sure the calculator is in degree mode. ( If you have your own TI calculator, see next slide for TI instructions on mode) If you do not have your own Ti or are not using the online resource, check your manual.

How to set the mode on the TI-Nspire After opening a calculator document Press the home screen button Select 5: Settings Then select Degree under the drop down for Angle and tab down and select OK Reopen the calculator Document by selecting 4: Current Then Select 2: Document Settings This step may be a little different depending on your operating system.

Sine Function For example to evaluate sin 40°… Make sure the calculator is in degree mode. ( If you have your own TI calculator, see next slide for TI instructions on mode) Then press the trig key and choose sin. (some calculators may just have a sin key) Press 40 and then enter It should show a result of 0. 642787… Note: If this did not work on your calculator, try typing-in 40 and then press the sin key. Press the = key to get the answer. There are several things that can go wrong and this can be tricky! Be sure to practice with a calculator to make sure you are getting the correct answers.

Sine Function Try each of these on your calculator: sin 55° 0. 819 sin 10° 0. 174 sin 87° 0. 999

Cosine Function Use your calculator to determine cos 50° First, press the trig key… …then choose cos. ( some calculators may have just a cos key) You should get an answer of 0. 642787. . . Note: If this did not work on your calculator, try typing-in 50 and then press the cos key. Press the = key to get the answer. There are several things that can go wrong and this can be tricky! Be sure to practice with a calculator to make sure you are getting the correct answers.

Cosine Function Try these on your calculator: cos 25° cos 0° cos 90° cos 45°

Cosine Function Try these on your calculator: n cos 25° = 0. 906 n cos 0° = 1 n cos 90° = 0 n cos 45° = 0. 707 n

Tangent Function Use your calculator to determine tan 40° n First, press the trig key… n …then choose tan. ( some calculators may just have a tan key. ) n You should get an answer of 0. 839. . . n n Note: If this did not work on your calculator, try typing-in 40 and then press the tan key. Press the = key to get the answer. There are several things that can go wrong and this can be tricky! Be sure to practice with a calculator to make sure you are getting the correct answers.

Tangent Function Try these on your calculator: n tan 5° = 0. 087 n tan 30° = 0. 577 n tan 80° = 5. 671 n tan 85° = 11. 430 n

Right Triangle Trigonometry is based upon ratios of the sides of right triangles. The sides of the right triangle are: hyp θ Ø and the hypotenuse of the right triangle. opp adj

Sine function The sin function specifies these two sides of the triangle, and they must be arranged as shown. hypotenuse opposite

Cosine function Cosine Function n The next trig function you need to know is the cosine function (cos): hypotenuse adjacent

Tangent function Function n The last trig function you need to know is the tangent function (tan): opposite adjacent

hypotenuse opposite

hypotenuse adjacent

opposite adjacent

We need a way to remember all of these ratios…

DISCLAIMER: I do not encourage or support the use of illegal drugs outside of mnemonic devices!

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SOHCAHTOA Old Hippie S in Opp Hyp Cos Adj Hyp Tan Opp Adj

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Finding sin, cos, and tan

SOHCAHTOA 10 8 6

Find the sine, the cosine, and the tangent of angle A B Give a fraction and decimal answer (round to 4 decimal places). 24. 5 8. 2 A 23. 1

$Find the sine, the cosine, and the tangent of angle A. Give a fraction$

Find the sine, the cosine, and the tangent of angle A. Give a fraction and decimal answer (round to 4 places). 10. 8 9 A 6

Find the values of the three trigonometric functions of . ? 5 4 3 Pythagorean Theorem: (3)² + (4)² = c² 5=c

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Finding a side

Find k 6. k 12 cm 25 o sin A = sin 25 = We have been given the hypotenuse of the right triangle and need to find the leg (k) opposite the 25 o angle so we use SINE: Sin A =

Finding a side from a triangle 4. 7 cm k We have been given the hypotenuse and need to find the leg (k) adjacent to the 30 o angle so we use 30 o Cos A = Cos 30 = COSINE:

We have been given the leg adjacent to the 50 o angle and need to find the leg opposite the 50 o angle so we use TAN: 5. 50 o 4 cm r Tan A = Tan 50 = Tan A =

Applications Involving Right Triangles A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78. 3. How tall is the Washington Monument? Solution: where x = 115 and y is the height of the monument. So, the height of the Washington Monument is y = x tan 78. 3 115(4. 82882) 555 feet.

Ex. 5 A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge? cos 60° x (cos 60°) = 200 60° x x

Ex. A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71. 5°. How tall is the tree? tan 71. 5° ? tan 71. 5° 50 y = 50 (tan 71. 5°) y = 50 (2. 98868)

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Using ratios to find angles Trig functions can also be used in reverse to find an angle from a ratio. To do this we use the sin-1, -1 cos and -1 tan function keys.

Inverse Sine Function Using sin-1 (inverse sin): If then Solve 0. 7315 = sin θ sin-1 (0. 7315) = θ for θ if sin θ = 0. 2419

Inverse Cosine Function Using cos-1 (inverse cosine): If then Solve 0. 9272 = cos θ cos-1 (0. 9272) = θ for θ if cos θ 0. 5150

Inverse Tangent Function Using tan-1 (inverse tangent): If then Solve 0. 5543 = tan θ tan-1 (0. 5543) = θ for θ if tan θ = 28. 64

Finding an angle from a triangle To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle. We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio. 1. Find angle C 14 cm 6 cm C a) Identify/label the names of the sides. b) Choose the ratio that contains BOTH of the letters.

3. 10 cm 12 cm y sin A = sin x = Given opp and hyp need to use sin: Sin A =

1. h 14 cm 6 cm a We have been given the adjacent and hypotenuse so we use COSINE: Cos A = Cos C = 0. 4286

2. Find angle x 3 cma x o 8 cm Given adj and opp need to use tan: Tan A = Tan x = 2. 6667

Try these 1. x 30 o 5 cm 2. 50 o 4 cm 3. 10 cm r 12 cm y Check your answers on the next slide

1. Cos A = x Cos 30 = x = 30 o 5 cm 2. Tan A = 50 o Tan 50 = 4 cm r 3. 10 cm sin A = 12 cm y sin y =

Review These are the only trig functions you will be using in this course. You need to memorize each one. Use the memory device: SOH CAH TOA

Review The sin function: sin A = hypotenuse opposite A

Review n The cosine function. cos A = hypotenuse A adjacent

Review n The tangent function. tan A = opposite A adjacent

Most Common Application: r θ x y

Review Solve for x: x = sin 30° x = cos 45° x = tan 20°

Review Solve for θ: 0. 7987 = sin θ 0. 9272 = cos θ 2. 145 = tan θ

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What if it’s not a right triangle? Begin here for April 13, 2020 lesson

Law of Sines -use when it is NOT a right triangle In order to use the law of sines you must know the length of a side opposite one of the given angles and one other measure.

When should the Law of Sines be used?

Law of Sines Example

Law of Sines Example - continued Begin by separating into two proportions. You must use the ratio that does not have a variable in both proportions Then cross multiply and solve each proportion Do not enter into the calculator until the very last step or you will be rounding multiple times and your margin of error will increase.