Download Holt McDougal Geometry

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. 11
3. If b = 6 and c = 10, find sin B. 0.6
Find AB.
Distance formula: √(x2 – x1)2+(y2 – y1)
4. A(8, 10), B(3, 0)
5. A(1, –2), B(2, 6)
Holt McDougal Geometry
Solving Right Triangles
Essential Question
• How can I use SOHCAHTOA to solve for
side lengths and angles?
Holt McDougal Geometry
Solving Right Triangles
Unit 2 Right triangles
• Section 4: Solving right triangles
Lesson 44
Holt McDougal Geometry
Solving Right Triangles
Standard(s):
• MCC9-12.G.SRT.8
Use trigonometric ratios and the
Pythagorean Theorem to solve right
triangles in applied problems
• MCC9-12.G.SRT.6
Understand that by similarity, side ratios in
right triangles are properties of the angles in
the triangle, leading to definitions of
trigonometric ratios for acute angles.
Holt McDougal Geometry
Solving Right Triangles
Example 3B: Calculating Trigonometric Ratios
Use your calculator to find the trigonometric
ratio. Round to the nearest hundredth.
cos 19°
cos 19°  0.95
Holt McDougal Geometry
Solving Right Triangles
Example 3C: Calculating Trigonometric Ratios
Use your calculator to find the trigonometric
ratio. Round to the nearest hundredth.
tan 65°
tan 65°  2.14
Holt McDougal Geometry
Solving Right Triangles
Example 4A: Using Trigonometric Ratios to Find
Lengths
Find the length. Round to
the nearest hundredth.
BC
is adjacent to the given angle, B. You are
given AC, which is opposite B. Since the
adjacent and opposite legs are involved, use a
tangent ratio.
Holt McDougal Geometry
Solving Right Triangles
Example 4A Continued
Write a trigonometric ratio.
Substitute the given values.
Multiply both sides by BC
and divide by tan 15°.
BC  38.07 ft
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Simplify the expression.
Solving Right Triangles
Example 4B: Using Trigonometric Ratios to Find
Lengths
Find the length. Round to
the nearest hundredth.
QR
is opposite to the given angle, P. You are
given PR, which is the hypotenuse. Since the
opposite side and hypotenuse are involved, use a
sine ratio.
Holt McDougal Geometry
Solving Right Triangles
Example 4B Continued
Write a trigonometric ratio.
Substitute the given values.
12.9(sin 63°) = QR
11.49 cm  QR
Holt McDougal Geometry
Multiply both sides by 12.9.
Simplify the expression.
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.
Holt McDougal Geometry
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°
Holt McDougal Geometry
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,
RT2 = RS2 + ST2
(5.7)2 = 52 + ST2
Since the acute angles of a right triangle are
complementary, mT  90° – 29°  61°.
Holt McDougal Geometry
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.
Holt McDougal Geometry
Solving Right Triangles
Example 4 Continued
Step 1 Find the side lengths. Plot points P, Q, and R.
PR = 7
Y
P
By the Distance Formula,
Q
X
R
Holt McDougal Geometry
PQ = 5
Solving Right Triangles
Example 4 Continued
Step 2 Find the angle measures.
Y
P
mP = 90°
Q
X
R
The acute s of a rt. ∆ are comp.
mR  90° – 54°  36°
Holt McDougal Geometry
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.
Holt McDougal Geometry
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.
C
38 ft
A
100 ft
B
Draw a right triangle to
represent the road.
A is the angle the road
makes with a horizontal line.
Holt McDougal Geometry