3 π 2 3 π 2
−135 ° −135 °
7 π 10 7 π 10
α = 150° α = 150°
β = 60° β = 60°
7 π 6 7 π 6
215 π 18 = 37.525 units 215 π 18 = 37.525 units
− 3 π 2 − 3 π 2 rad/s
1655 kilometers per hour
7.2 Right Triangle Trigonometry
sin t = 33 65 , cos t = 56 65 , tan t = 33 56 , sec t = 65 56 , csc t = 65 33 , cot t = 56 33 sin t = 33 65 , cos t = 56 65 , tan t = 33 56 , sec t = 65 56 , csc t = 65 33 , cot t = 56 33
sin ( π 4 ) = 2 2 , cos ( π 4 ) = 2 2 , tan ( π 4 ) = 1 , sec ( π 4 ) = 2 , csc ( π 4 ) = 2 , cot ( π 4 ) = 1 sin ( π 4 ) = 2 2 , cos ( π 4 ) = 2 2 , tan ( π 4 ) = 1 , sec ( π 4 ) = 2 , csc ( π 4 ) = 2 , cot ( π 4 ) = 1
adjacent = 10 ; opposite = 10 3 ; adjacent = 10 ; opposite = 10 3 ; missing angle is π 6 π 6
About 52 ft
7.3 Unit Circle
cos ( t ) = − 2 2 , sin ( t ) = 2 2 cos ( t ) = − 2 2 , sin ( t ) = 2 2
cos ( π ) = − 1 , sin ( π ) = 0 cos ( π ) = − 1 , sin ( π ) = 0
sin ( t ) = − 7 25 sin ( t ) = − 7 25
approximately 0.866025403
- ⓐ cos ( 315° ) = 2 2 , sin ( 315° ) = – 2 2 cos ( 315° ) = 2 2 , sin ( 315° ) = – 2 2
- ⓑ cos ( − π 6 ) = 3 2 , sin ( − π 6 ) = − 1 2 cos ( − π 6 ) = 3 2 , sin ( − π 6 ) = − 1 2
( 1 2 , − 3 2 ) ( 1 2 , − 3 2 )
7.4 The Other Trigonometric Functions
sin t = − 2 2 cos t = 2 2 , tan t = − 1 , s e c t = 2 , csc t = − 2 , cot t = − 1 sin t = − 2 2 cos t = 2 2 , tan t = − 1 , s e c t = 2 , csc t = − 2 , cot t = − 1
sin π 3 = 3 2 , cos π 3 = 1 2 , tan π 3 = 3 , s e c π 3 = 2 , c s c π 3 = 2 3 3 , c o t π 3 = 3 3 sin π 3 = 3 2 , cos π 3 = 1 2 , tan π 3 = 3 , s e c π 3 = 2 , c s c π 3 = 2 3 3 , c o t π 3 = 3 3
sin ( − 7 π 4 ) = 2 2 , cos ( − 7 π 4 ) = 2 2 , tan ( − 7 π 4 ) = 1 , sec ( − 7 π 4 ) = 2 , csc ( − 7 π 4 ) = 2 , cot ( − 7 π 4 ) = 1 sin ( − 7 π 4 ) = 2 2 , cos ( − 7 π 4 ) = 2 2 , tan ( − 7 π 4 ) = 1 , sec ( − 7 π 4 ) = 2 , csc ( − 7 π 4 ) = 2 , cot ( − 7 π 4 ) = 1
sin t sin t
cos t = − 8 17 , sin t = 15 17 , tan t = − 15 8 csc t = 17 15 , cot t = − 8 15 cos t = − 8 17 , sin t = 15 17 , tan t = − 15 8 csc t = 17 15 , cot t = − 8 15
sin t = − 1 , cos t = 0 , tan t = Undefined sec t = Undefined, csc t = − 1 , cot t = 0 sin t = − 1 , cos t = 0 , tan t = Undefined sec t = Undefined, csc t = − 1 , cot t = 0
sec t = 2 , csc t = 2 , tan t = 1 , cot t = 1 sec t = 2 , csc t = 2 , tan t = 1 , cot t = 1
≈ − 2.414 ≈ − 2.414
7.1 Section Exercises
Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.
Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.
4 π 3 4 π 3
2 π 3 2 π 3
7 π 2 ≈ 11.00 in 2 7 π 2 ≈ 11.00 in 2
81 π 20 ≈ 12.72 cm 2 81 π 20 ≈ 12.72 cm 2
π 2 π 2 radians
−3 π −3 π radians
π π radians
5 π 6 5 π 6 radians
5.02 π 3 ≈ 5.26 5.02 π 3 ≈ 5.26 miles
25 π 9 ≈ 8.73 25 π 9 ≈ 8.73 centimeters
21 π 10 ≈ 6.60 21 π 10 ≈ 6.60 meters
104.7198 cm 2
0.7697 in 2
8 π 9 8 π 9
1320 1320 rad/min 210.085 210.085 RPM
7 7 in./s, 4.77 RPM , 28.65 28.65 deg/s
1 , 809 , 557.37 mm/min = 1 , 809 , 557.37 mm/min = 30.16 m/s 30.16 m/s
5.76 5.76 miles
794 miles per hour
2,234 miles per hour
11.5 inches
7.2 Section Exercises
The tangent of an angle is the ratio of the opposite side to the adjacent side.
For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.
b = 20 3 3 , c = 40 3 3 b = 20 3 3 , c = 40 3 3
a = 10,000 , c = 10,00.5 a = 10,000 , c = 10,00.5
b = 5 3 3 , c = 10 3 3 b = 5 3 3 , c = 10 3 3
5 29 29 5 29 29
5 41 41 5 41 41
c = 14 , b = 7 3 c = 14 , b = 7 3
a = 15 , b = 15 a = 15 , b = 15
b = 9.9970 , c = 12.2041 b = 9.9970 , c = 12.2041
a = 2.0838 , b = 11.8177 a = 2.0838 , b = 11.8177
a = 55.9808 , c = 57.9555 a = 55.9808 , c = 57.9555
a = 46.6790 , b = 17.9184 a = 46.6790 , b = 17.9184
a = 16.4662 , c = 16.8341 a = 16.4662 , c = 16.8341
498.3471 ft
22.6506 ft
368.7633 ft
7.3 Section Exercises
The unit circle is a circle of radius 1 centered at the origin.
Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, t , t , formed by the terminal side of the angle t t and the horizontal axis.
The sine values are equal.
60° , 60° , Quadrant IV, sin ( 300° ) = − 3 2 sin ( 300° ) = − 3 2 , cos ( 300° ) = 1 2 cos ( 300° ) = 1 2
45° , 45° , Quadrant II, sin ( 135° ) = 2 2 sin ( 135° ) = 2 2 , cos ( 135° ) = − 2 2 cos ( 135° ) = − 2 2
60° , 60° , Quadrant II, sin ( 120° ) = 3 2 sin ( 120° ) = 3 2 , cos ( 120° ) = − 1 2 cos ( 120° ) = − 1 2
30° , 30° , Quadrant II, sin ( 150° ) = 1 2 sin ( 150° ) = 1 2 , cos ( 150° ) = − 3 2 cos ( 150° ) = − 3 2
π 6 , π 6 , Quadrant III, sin ( 7 π 6 ) = − 1 2 sin ( 7 π 6 ) = − 1 2 , cos ( 7 π 6 ) = − 3 2 cos ( 7 π 6 ) = − 3 2
π 4 , π 4 , Quadrant II, sin ( 3 π 4 ) = 2 2 sin ( 3 π 4 ) = 2 2 , cos ( 4 π 3 ) = − 2 2 cos ( 4 π 3 ) = − 2 2
π 3 , π 3 , Quadrant II, sin ( 2 π 3 ) = 3 2 sin ( 2 π 3 ) = 3 2 , cos ( 2 π 3 ) = − 1 2 cos ( 2 π 3 ) = − 1 2
π 4 , π 4 , Quadrant IV, sin ( 7 π 4 ) = − 2 2 , cos ( 7 π 4 ) = 2 2 sin ( 7 π 4 ) = − 2 2 , cos ( 7 π 4 ) = 2 2
− 15 4 − 15 4
( −10 , 10 3 ) ( −10 , 10 3 )
( –2.778 , 15.757 ) ( –2.778 , 15.757 )
[ –1 , 1 ] [ –1 , 1 ]
sin t = 1 2 , cos t = − 3 2 sin t = 1 2 , cos t = − 3 2
sin t = − 2 2 , cos t = − 2 2 sin t = − 2 2 , cos t = − 2 2
sin t = 3 2 , cos t = − 1 2 sin t = 3 2 , cos t = − 1 2
sin t = − 2 2 , cos t = 2 2 sin t = − 2 2 , cos t = 2 2
sin t = 0 , cos t = − 1 sin t = 0 , cos t = − 1
sin t = − 0.596 , cos t = 0.803 sin t = − 0.596 , cos t = 0.803
sin t = 1 2 , cos t = 3 2 sin t = 1 2 , cos t = 3 2
sin t = − 1 2 , cos t = 3 2 sin t = − 1 2 , cos t = 3 2
sin t = 0.761 , cos t = − 0.649 sin t = 0.761 , cos t = − 0.649
sin t = 1 , cos t = 0 sin t = 1 , cos t = 0
− 6 4 − 6 4
( 0 , –1 ) ( 0 , –1 )
37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds
7.4 Section Exercises
Yes, when the reference angle is π 4 π 4 and the terminal side of the angle is in quadrants I and III. Thus, a x = π 4 , 5 π 4 , x = π 4 , 5 π 4 , the sine and cosine values are equal.
Substitute the sine of the angle in for y y in the Pythagorean Theorem x 2 + y 2 = 1. x 2 + y 2 = 1. Solve for x x and take the negative solution.
The outputs of tangent and cotangent will repeat every π π units.
2 3 3 2 3 3
− 2 3 3 − 2 3 3
− 3 3 − 3 3
sin t = − 2 2 3 sin t = − 2 2 3 , sec t = − 3 sec t = − 3 , csc t = − 3 2 4 csc t = − 3 2 4 , tan t = 2 2 tan t = 2 2 , cot t = 2 4 cot t = 2 4
sec t = 2 , sec t = 2 , csc t = 2 3 3 , csc t = 2 3 3 , tan t = 3 , tan t = 3 , cot t = 3 3 cot t = 3 3
− 2 2 − 2 2
sin t = 2 2 sin t = 2 2 , cos t = 2 2 cos t = 2 2 , tan t = 1 tan t = 1 , cot t = 1 cot t = 1 , sec t = 2 sec t = 2 , csc t = 2 csc t = 2
sin t = − 3 2 sin t = − 3 2 , cos t = − 1 2 cos t = − 1 2 tan t = 3 tan t = 3 , cot t = 3 3 cot t = 3 3 , sec t = − 2 sec t = − 2 , csc t = − 2 3 3 csc t = − 2 3 3
sin ( t ) ≈ 0.79 sin ( t ) ≈ 0.79
csc t ≈ 1.16 csc t ≈ 1.16
sin t cos t = tan t sin t cos t = tan t
13.77 hours, period: 1000 π 1000 π
3.46 inches
Review Exercises
− 7 π 6 − 7 π 6
10.385 meters
2 π 11 2 π 11
1036.73 miles per hour
a = 10 3 , c = 2 106 3 a = 10 3 , c = 2 106 3
a = 5 3 2 , b = 5 2 a = 5 3 2 , b = 5 2
369.2136 ft
all real numbers
cosine, secant
Practice Test
6.283 centimeters
3.351 feet per second, 2 π 75 2 π 75 radians per second
a = 9 2 , b = 9 3 2 a = 9 2 , b = 9 3 2
real numbers
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Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites
- Authors: Jay Abramson
- Publisher/website: OpenStax
- Book title: Algebra and Trigonometry
- Publication date: Feb 13, 2015
- Location: Houston, Texas
- Book URL: https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites
- Section URL: https://openstax.org/books/algebra-and-trigonometry/pages/chapter-7
© Dec 8, 2021 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.
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1 2-7.71 peri me-er- - 103. BYS 32 m Directions: Given the side lengths, determine whether the triangle is acute. right, obtuse, or not a triangle. 13. 10, 24, 26 2 Seq O NotaA Acute 12. 20, 23, 41 + > q 201 14. 16. 6. 13, 20 a NotaA a Acute a Right Obtuse O Acute O Right a Obtuse a NotaA Y' Acute a Right Obtuse a a a a Right Obtuse Not a A ...
Unit 8 - Right Triangles & Trigonometry. Directions: Use the Law of Cosines to solve for x. Round your answer to the nearest tenth. - - = 8105, 121 = cosx COS X cosx 2q{u -2.0 18 2.1131 46. A utility pole is supported by two wires, one on each side going in the opposite direction. The two wires form a 75' angle at the utility pole.
Question: Name: Unit 8: Right Triangles & Trigonometry Homework 7: Law of Sines Per Date: ** This is a 2-page document! Directions: Use the Law of Sines to find each missing side or angle. Round to the nearest tenth. 1. 2. 22 5 65 46 29 53 3. т 73 59 1280 18 12 15 5. 191 75 32 26 28 7. 9 514 52 70 16 Gna Wilson ( Ang Algebra C, 2014-2018
1.elevation2.depression. 3. ∠1 is formed by a horiz. line and a line of sight to a pt. above the line. It is an ∠of elevation. 4. ∠2 is formed by a horiz. line and a line of sight to a pt. below the line. It is an ∠of depression. 5. ∠3 is formed by a horiz. line and a line of sight to a pt. above the line.
Identify if the triangle is a right triangle or not. 20, 48, 52 By the converse of Pythagorean theorem, check the sum of squares of smaller sides with the square of largest side i.e., 220+482=400+2304=2704 252=2704 → 202+482= 522 The triangle is a right triangle. 3. The longest side in a right triangle is: e. hypotenuse f. adjacent g. opposite h.
Add-on. U08.AO.01 - Terminology Warm-Up for the Trigonometric Ratios (Before Lesson 2) RESOURCE. ANSWER KEY. EDITABLE RESOURCE. EDITABLE KEY.
The diagram below shows displays triangle ABC. a) Find the missing side to the nearest tenth. b) Find the perimeter of triangle ABC. 16. Find the missing side of the following triangle. Round your answer to the nearest hundredth. 17. Use the diagram to the right to answer the questions below. (a) Calculate the length of AC.
x=. 35.6. Solve for the missing angle x, round to the nearest tenth. 41.2. Solve for the missing angle x, round to the nearest tenth. 43.9. Solve for the missing angle x, round to the nearest tenth. Unit 8: Trigonometry REVIEW. 2.5.
Unit 8: right triangles & trigonometry homework 4 trigonometry finding sides and angles. verified. Verified answer. star. 5 /5. 1. Answer: 9= 71.67° 10= 60.65° 11= 86.59° 12= 62.30° 13= 34.51° 14= 51.71° 15= 22.87° 16= 44.63° Step-by-step explanation: The law of sine requires that if we ha….
Unit 8 Test: Right Triangles & Trigonometry. Geometric Mean formula. Click the card to flip 👆. a/x = x/b. Click the card to flip 👆. 1 / 10.
Step 1. (1) Observe the given triangle very carefully. Directions: Use the Law of Sines to find each missing side or angle. Round to the nearest tenth. 2. 1. x 22 659 5 46 29 53 3. 128 73 599 12 18 15 5. 6. X 191 781 78 32 28 7. 8.
Terms in this set (16) A triangle has side lengths of 34 in, 20 in, and 47 in. Is the triangle acute, obtuse or right? B. Obtuse. In triangle ABC, A is a right angle, and M B=45 degrees. A. 14_/2 ft. Quilt squares are cut on the diagonal to form triangular whilt pieces.
Geometry questions and answers. Name: Cayce Date: Per: Unit 8: Right Triangles & Trigonometry Homework 4: Trigonometric Ratios & Finding Missing Sides SOH CAH TOA ** This is a 2-page document! ** 1. 48/50 Р sin R = Directions: Give each trig ratio as a fraction in simplest form. 14/50 48 sin Q = 48150 cos 14/48 tan Q = Q 14150 14 .
Introduction to Further Applications of Trigonometry; 10.1 Non-right Triangles: Law of Sines; 10.2 Non-right Triangles: Law of Cosines; 10.3 Polar Coordinates; 10.4 Polar Coordinates: Graphs; 10.5 Polar Form of Complex Numbers; 10.6 Parametric Equations; 10.7 Parametric Equations: Graphs; 10.8 Vectors
Unit 8 Part 1 - Pythagorean Triples, Pythagorean Theorem and its Converse, Special Right Triangles ... Special right Triangles Geometry B Unit 4. Teacher 5 terms. helphander. Preview. gerometry b unit 4 lesson 1 the pythagorean theorem and its converse. 5 terms. dingerbugal1993. Preview. Chapter 2 Key Vocabulary. 18 terms. kmcraosmun25. Preview ...
This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Question: Unit 8: Right Triangles & Trigonometry Homework 5: Trigonometry: Finding Sides and Angles ge document! ** enth. 2. TAN 39 - = 27 399 X х 27 X=27 Tana9 X=33.34 159 33 х 6. 5 27° х.
The main trigonometric ratios are presented below. Triangle 1. For angle D you will find: For angle E you will find: Triangle 2. The question gives an angle (62°) and the adjacent side (25) from the angle 62° of the right triangle. Therefore, you can find x from the trigonometric ratio of tan (62°): Triangle 3.
trigonometry. the study of the relationship between side lengths and angles in triangles. opposite leg. the leg across from a given acute angle in a right triangle. adjacent leg. the leg that touches a given acute angle in a right triangle. theta. the symbol θ used as a variable for an angle. sine/sin.
Final answer: In trigonometry, similar right triangles have proportional corresponding sides. To find the geometric mean of two values, set up a proportion using the corresponding sides of two similar triangles. Explanation: In trigonometry, similar right triangles are triangles that have the same shape but may be different sizes.
Geometry. Geometry questions and answers. Name: Date: Unit 8: Right Triangles & Trigonometry Homework 9: Law of Sines & Law of Cosines; + Applications ** This is a 2-page document ** Per Directions: Use the Law of Sines and/or the Law of Cosines to solve each triangle. Round to the nearest tenth when necessary 1. OR 19 mZP P 85 13 R MZO - 2.
c2>a2+b2. Right Triangle. c^2 = a^2 + b^2. angle of elevation. angle formed by a horizontal line and a line of sight to a point above the line. angle of depression. angle formed by a horizontal line and a line of sight to a point below the line. Study with Quizlet and memorize flashcards containing terms like Pythagorean Theorem, Converse of ...
See Answer. Question: Name: Unit 7: Right Triangles & Trigonometry Date: Per Homework 3: Similar Right Triangles & Geometric Mean ** This is a 2-page document Directions: Identity the similar triangles in the diagram, then sketch them so the corresponding sides and angles have the same orientation 1. 2. Directions Solve for 29 10 20 21 6.
Pythagorean Theorem. In the case of a right triangle, a²+b²=c². Converse of the Pythagorean Theorem. If the angles are summative in terms of a²+b²=c², it is a right triangle. Pythagorean Triple. Three integers that, as side lengths of a triangle, form a right triangle (Ex. 3/4/5 or 5/12/13) 3-4-5. Pythagorean Triple.