This document discusses solving right triangles using trigonometric ratios. It includes examples of identifying angles from trigonometric ratios, calculating angle measures from ratios using inverse functions, solving right triangles when given side lengths or an angle measure, and solving triangles in the coordinate plane. One example problem solves for the angle of a road given its grade in percentage form.
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One of the best known mathematical formulas is Pythagorean Theorem,Over 2000 years ago there was an amazing discovery about triangles:
When a triangle has a right angle (90°) and squares are made on each of the three sides,then the biggest square has the exact same area as the other two squares put together! Maths is Fun
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A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
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This will be used as part of your Personal Professional Portfolio once graded.
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
1. Holt Geometry
8-3 Solving Right Triangles8-3 Solving Right Triangles
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
2. Holt Geometry
8-3 Solving Right Triangles
Warm Up
Use ∆ABC for Exercises 1–3.
1. If a = 8 and b = 5, find c.
2. If a = 60 and c = 61, find b.
3. If b = 6 and c = 10, find sin B.
Find AB.
4. A(8, 10), B(3, 0)
5. A(1, –2), B(2, 6)
11
0.6
3. Holt Geometry
8-3 Solving Right Triangles
Use trigonometric ratios to find angle
measures in right triangles and to solve
real-world problems.
Objective
4. Holt Geometry
8-3 Solving Right Triangles
San Francisco, California, is
famous for its steep streets. The
steepness of a road is often
expressed as a percent grade.
Filbert Street, the steepest street
in San Francisco, has a 31.5%
grade. This means the road rises
31.5 ft over a horizontal distance
of 100 ft, which is equivalent to a
17.5° angle. You can use
trigonometric ratios to change a
percent grade to an angle
measure.
5. Holt Geometry
8-3 Solving Right Triangles
Example 1: Identifying Angles from Trigonometric
Ratios
Since cos A = cos∠2, ∠2 is ∠A.
Use the trigonometric
ratio to
determine which angle
of the triangle is ∠A.
Cosine is the ratio of the adjacent
leg to the hypotenuse.
The leg adjacent to ∠1 is 1.4. The
hypotenuse is 5.
The leg adjacent to ∠2 is 4.8. The
hypotenuse is 5.
6. Holt Geometry
8-3 Solving Right Triangles
Check It Out! Example 1a
Use the given trigonometric
ratio to determine which
angle of the triangle is ∠A.
Sine is the ratio of the opposite
leg to the hypotenuse.
The leg adjacent to ∠1 is 27. The
hypotenuse is 30.6.
The leg adjacent to ∠2 is 14.4.
The hypotenuse is 30.6.
Since sin∠A = sin∠2, ∠2 is ∠A.
7. Holt Geometry
8-3 Solving Right Triangles
Check It Out! Example 1b
Use the given trigonometric
ratio to determine which
angle of the triangle is ∠A.
tan A = 1.875
Tangent is the ratio of the
opposite leg to the adjacent leg.
The leg opposite to ∠1 is 27. The
leg adjacent is 14.4.
The leg opposite to ∠2 is 14.4.
The leg adjacent is 27.
Since tan∠A = tan∠1, ∠1 is ∠A.
8. Holt Geometry
8-3 Solving Right Triangles
In Lesson 8-2, you learned that sin 30° = 0.5.
Conversely, if you know that the sine of an acute
angle is 0.5, you can conclude that the angle
measures 30°. This is written as sin-1
(0.5) = 30°.
9. Holt Geometry
8-3 Solving Right Triangles
If you know the sine, cosine, or tangent of an acute
angle measure, you can use the inverse
trigonometric functions to find the measure of the
angle.
10. Holt Geometry
8-3 Solving Right Triangles
Example 2: Calculating Angle Measures from
Trigonometric Ratios
Use your calculator to find each angle measure to
the nearest degree.
A. cos-1
(0.87) B. sin-1
(0.85) C. tan-1
(0.71)
cos-1
(0.87) ≈ 30° sin-1
(0.85) ≈ 58° tan-1
(0.71) ≈ 35°
11. Holt Geometry
8-3 Solving Right Triangles
Check It Out! Example 2
Use your calculator to find each angle measure
to the nearest degree.
a. tan-1
(0.75) b. cos-1
(0.05) c. sin-1
(0.67)
cos-1
(0.05) ≈ 87° sin-1
(0.67) ≈ 42°tan-1
(0.75) ≈ 35°
12. Holt Geometry
8-3 Solving Right Triangles
Using given measures to find the unknown angle
measures or side lengths of a triangle is known as
solving a triangle. To solve a right triangle, you need
to know two side lengths or one side length and an
acute angle measure.
13. Holt Geometry
8-3 Solving Right Triangles
Example 3: Solving Right Triangles
Find the unknown measures.
Round lengths to the nearest
hundredth and angle measures to
the nearest degree.
Method 1: By the Pythagorean Theorem,
Since the acute angles of a right triangle are
complementary, m∠T ≈ 90° – 29° ≈ 61°.
RT2
= RS2
+ ST2
(5.7)2
= 52
+ ST2
14. Holt Geometry
8-3 Solving Right Triangles
Example 3 Continued
Method 2:
Since the acute angles of a right triangle are
complementary, m∠T ≈ 90° – 29° ≈ 61°.
, so ST = 5.7 sinR.
15. Holt Geometry
8-3 Solving Right Triangles
Check It Out! Example 3
Find the unknown measures.
Round lengths to the nearest
hundredth and angle measures
to the nearest degree.
Since the acute angles of a right triangle are
complementary, m∠D = 90° – 58° = 32°.
, so EF = 14 tan 32°. EF ≈ 8.75
DF2
= 142
+ 8.752
DF2
= ED2
+ EF2
DF ≈ 16.51
16. Holt Geometry
8-3 Solving Right Triangles
Example 4: Solving a Right Triangle in the Coordinate
Plane
The coordinates of the vertices of ∆PQR are
P(–3, 3), Q(2, 3), and R(–3, –4). Find the side
lengths to the nearest hundredth and the
angle measures to the nearest degree.
17. Holt Geometry
8-3 Solving Right Triangles
Example 4 Continued
Step 1 Find the side lengths. Plot points P, Q, and R.
P Q
R
Y
X
By the Distance Formula,
PR = 7 PQ = 5
18. Holt Geometry
8-3 Solving Right Triangles
Example 4 Continued
Step 2 Find the angle measures.
P Q
R
Y
X
m∠P = 90°
m∠R ≈ 90° – 54° ≈ 36°
The acute ∠s of a rt. ∆ are comp.
19. Holt Geometry
8-3 Solving Right Triangles
Check It Out! Example 4
The coordinates of the vertices of ∆RST are
R(–3, 5), S(4, 5), and T(4, –2). Find the side
lengths to the nearest hundredth and the
angle measures to the nearest degree.
20. Holt Geometry
8-3 Solving Right Triangles
Check It Out! Example 4 Continued
Step 1 Find the side lengths. Plot points R, S, and T.
R S
T
Y
X
RS = ST = 7
By the Distance Formula,
21. Holt Geometry
8-3 Solving Right Triangles
Check It Out! Example 4 Continued
Step 2 Find the angle measures.
m∠S = 90°
m∠R ≈ 90° – 45° ≈ 45° The acute ∠s of a rt. ∆ are comp.
22. Holt Geometry
8-3 Solving Right Triangles
Example 5: Travel Application
A highway sign warns that a section of road
ahead has a 7% grade. To the nearest degree,
what angle does the road make with a
horizontal line?
Change the percent grade to a fraction.
A 7% grade means the road rises (or falls) 7 ft for
every 100 ft of horizontal distance.
Draw a right triangle to
represent the road.
∠A is the angle the road
makes with a horizontal line.
23. Holt Geometry
8-3 Solving Right Triangles
Check It Out! Example 5
Baldwin St. in Dunedin, New Zealand, is the
steepest street in the world. It has a grade of
38%. To the nearest degree, what angle does
Baldwin St. make with a horizontal line?
Change the percent
grade to a fraction.
A 38% grade means the road rises (or falls) 38 ft for
every 100 ft of horizontal distance.
Draw a right triangle to
represent the road.
∠A is the angle the road
makes with a horizontal line.
100 ft
38 ft
A B
C
24. Holt Geometry
8-3 Solving Right Triangles
Lesson Quiz: Part I
Use your calculator to find each angle
measure to the nearest degree.
1. cos-1
(0.97)
2. tan-1
(2)
3. sin-1
(0.59)
14°
63°
36°
25. Holt Geometry
8-3 Solving Right Triangles
Lesson Quiz: Part II
Find the unknown measures. Round lengths
to the nearest hundredth and angle
measures to the nearest degree.
4. 5.
DF ≈ 5.7; m∠D ≈ 68°;
m∠F ≈ 22°
AC ≈ 0.63; BC ≈ 2.37;
m ∠B = 15°
26. Holt Geometry
8-3 Solving Right Triangles
Lesson Quiz: Part III
6. The coordinates of the vertices of ∆MNP are
M (–3, –2), N(–3, 5), and P(6, 5). Find the
side lengths to the nearest hundredth and the
angle measures to the nearest degree.
MN = 7; NP = 9; MP ≈ 11.40; m∠N = 90°;
m∠M ≈ 52°; m∠P ≈ 38°