Trigonometry Basics Right Triangle Trigonometry 1 slide Greek

Trigonometry Basics Right Triangle Trigonometry 1 slide

Greek Letters are used in Trigonometry to represent unknown angle measures. They are used as “variables” when looking for the measure of an angle in the same way we use x, y, z, etc. in geometry to represent unknown segment lengths. 2 slide

Greek Letter Pronounced “theta” Represents an unknown angle 3 slide

Pronounced “alpha” Represents an unknown angle 4 slide

Pronounced “beta” Represents an unknown angle 5 slide

Pronounced “sigma” Represents an unknown angle 6 slide

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The Trigonometric Functions we will be looking at SINE COSINE TANGENT 8 slide

The Trigonometric Functions SINE COSINE TANGENT 9 slide

SINE Prounounced “sign” 10 slide

COSINE Prounounced “co-sign” 11 slide

TANGENT Prounounced “tan-gent” 12 slide

Finding the ratios The simplest form of question is finding the decimal value of the ratio of a given angle. They will look similar to this. Find using calculator: 1) sin 30 = 2) cos 23 = 3) tan 78 = 4) tan 27 = 5) sin 68 = 13 slide

Sine Function For example to evaluate sin 40°… Make sure the calculator is in degree 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. 14 slide

Sine Function Try each of these on your calculator: sin 55° sin 10° sin 87° 15 slide

Sine Function Try each of these on your calculator: n sin 55° = 0. 819 n sin 10° = 0. 174 n sin 87° = 0. 999 n 16 slide

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. 17 slide

Cosine Function Try these on your calculator: cos 25° cos 0° cos 90° cos 45° 18 slide

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 19 slide

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. 20 slide

Tangent Function Try these on your calculator: n tan 5° n tan 30° n tan 85° n 21 slide

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 22 slide

Right Triangle Trigonometry is based upon ratios of the sides of right triangles. The sides of the right triangle are: hyp θ opp adj Ø and the hypotenuse of the right triangle. 23 slide

Module 2 – Lesson 25 p. 164 -165 (#1 -6)

Sine function The sin function specifies these two sides of the triangle, and they must be arranged as shown. hypotenuse opposite 24 slide

Cosine function Cosine Function n The next trig function you need to know is the cosine function (cos): hypotenuse adjacent 25 slide

Tangent function Function n The last trig function you need to know is the tangent function (tan): opposite adjacent 26 slide

hypotenuse opposite 27 slide

hypotenuse adjacent 28 slide

opposite adjacent 29 slide

We need a way to remember all of these ratios… 30 slide

DISCLAIMER: I do not encourage or support the use of illegal drugs outside of mnemonic devices! 31 slide

Some Old Hippies Caught Another Hippie Trippin’ On Old Hippies Acid BUSTED! 32 slide

SOHCAHTOA Old Hippie S in Opp Hyp Cos Adj Hyp Tan Opp Adj 33 slide

SOHCAHTOA style 34 slide

Finding sin, cos, and tan 35 slide

SOHCAHTOA 10 8 6 36 slide

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 37 slide

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 41 slide

Find the values of the three trigonometric functions of . ? 5 4 Pythagorean Theorem: (3)² + (4)² = c² 5=c 3 42 slide

Complete Module 2 - Lesson 26 p. 174 (#1 -3)

Finding a side 43 slide

6. k 12 cm 25 o We have been given the opp and hyp so we use SINE: Sin A = sin 25 = 12(sin 25) = k 5. 1 cm = k 38 slide

Finding a side from a triangle 4. 7 cm k 30 o We have been given the adj and hyp so we use COSINE: Cos A = Cos 30 = 7(Cos 30) = k 6. 1 cm = k 39 slide

We have been given the opp and adj so we use TAN: 5. 50 o 4 cm Tan A = r tan A = tan 50 = 4(tan 50 )= r 4. 8 cm = r 40 slide

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. 44 slide

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 X = 400 yards 45 slide

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? Opp tan 71. 5°= Adj ? tan 71. 5° 50 y = 50 (tan 71. 5°) y = 50 (2. 98868) 46 slide

END OF LESSON 1 Complete Module 2 - Lesson 25 pp. 167 ( Exercise #10) Complete Module 2 -Lesson 26 pp. 178 -181 (Problem Set # 1, 4, 6 -9) Quiz tomorrow over trig functions Take notes on the remaining slides of this power point. 47 slide

Lesson 2 Using Trig Ratios to Find Angle Measures

Using ratios to find angles Trig ratios 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 48 slide

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 49 slide

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 50 slide

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 51 slide

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. 52 slide

3. 10 cm Given opp and hyp need to use sin: 12 cm Sin sin = 0. 8333 = = sin-1 (0. 8333) = 56. 4 o 53 slide

1. We have been given the adjacent and hypotenuse so we use COSINE: h 14 cm Cos = 6 cm a Cos = = 64. 6 o 54 slide

2. Find angle x 3 cma Given adj and opp need to use tan: x Tan A = o 8 cm Tan A = Tan x = 2. 6667 x = tan-1 (2. 6667) x = 69. 4 o 55 slide

1. Cos A = x Cos 30 = x = 30 o 5 cm x = 5. 8 cm 2. Tan A = 50 o Tan 50 = 4 cm Tan 50 x 4 = r 4. 8 cm = r r 3. 10 cm sin A = 12 cm y sin y = 0. 8333 y = sin-1 (0. 8333) y = 56. 4 o 56 slide

Fancy Parody - Soh. Cah. Toa https: //youtu. be/w. Rsl. Pd. Ugg. HY

Get Triggy With It 57 slide

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 58 slide

Review The sin function: sin A = hypotenuse opposite A 59 slide

Review n The cosine function. cos A = hypotenuse A adjacent 60 slide

Review n The tangent function. tan A = opposite A adjacent 61 slide

Most Common Application: r θ y x 62 slide

Review Solve for x: x = sin 30° x = cos 45° x = tan 20° 63 slide

Review Solve for θ: 0. 7987 = sin θ 0. 9272 = cos θ 2. 145 = tan θ 64 slide

What if it’s not a right triangle? - Use the Law of Cosines: The Law of Cosines In any triangle ABC, with sides a, b, and c, 65 slide

What if it’s not a right triangle? Law of Cosines - The square of the magnitude of the resultant vector is equal to the sum of the magnitude of the squares of the two vectors, minus two times the product of the magnitudes of the vectors, multiplied by the cosine of the angle between them. R 2 = A 2 + B 2 – 2 AB cosθ θ 66 slide

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