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Right Triangle Trigonometry
The Pythagorean Theorem
The Pythagorean Theorem states that the following relationship exists among the lengths of the legs, a and b, and the length of the hypotenuse, c, of any right triangle.
[pic] a2 + b2 ’ c2
Use the Pythagorean Theorem to find the value of x in each triangle.
[pic] [pic]
a2 + b2 ’ c2 Pythagorean Theorem a2 + b2 ’ c2
x2 + 62 ’ 92 Substitute. x2 + 42 ’ (x + 2)2
x2 + 36 ’ 81 Take the squares. x2 + 16 ’ x2 + 4x + 4
x2 ’ 45 Simplify. 4x ’ 12
x ’ [pic] x ’ 3
x ’ [pic]
Find the value of x. Give your answer in simplest radical form.
1. [pic] 2. [pic]
3. [pic] 4. [pic]
Reteach
The Pythagorean Theorem continued
A Pythagorean triple is a set of three
nonzero whole numbers a, b, and c
that satisfy the equation a2 + b2 ’ c2.
You can use the following theorem to classify triangles by their angles if you know their side lengths. Always use the length of the longest side for c.
Consider the measures 2, 5, and 6. They can be the side lengths of a triangle since
2 + 5 > 6, 2 + 6 > 5, and 5 + 6 > 2. If you substitute the values into c2 < a2 + b2, you
get 36 > 29. Since c2 > a2 + b2, a triangle with side lengths 2, 5, and 6 must be obtuse.
Find the missing side length. Tell whether the side lengths form a Pythagorean triple. Explain.
5. [pic] 6. [pic]
Tell whether the measures can be the side lengths of a triangle.
If so, classify the triangle as acute, obtuse, or right.
7. 4, 7, 9 8. 10, 13, 16 9. 8, 8, 11
10. 9, 12, 15 11. 5, 14, 20 12. 4.5, 6, 10.2
Applying Special Right Triangles
In a 450-450-900triangle, if a leg
length is x, then the hypotenuse
length is [pic]
Use the 450-450-900 Triangle Theorem to find the value of x in ΔEFG.
Every isosceles right triangle is a 450-450-900 triangle. Triangle
EFG is a 458-458-908 triangle with a hypotenuse of length 10.
[pic] Hypotenuse is [pic] times the length of a leg.
[pic] Divide both sides by [pic]
[pic] Rationalize the denominator.
Find the value of x. Give your answers in simplest radical form.
1. [pic] 2. [pic]
3. [pic] 4. [pic]
Applying Special Right Triangles
In a 300-600-900 triangle, if the shorter leg
length is x, then the hypotenuse length
is 2x and the longer leg length is x.
Use the 300-600-900 Triangle Theorem to find the values
of x and y in ΔHJK.
[pic] Longer leg ’ shorter leg multiplied by [pic]
[pic] Divide both sides by [pic]
[pic] Rationalize the denominator.
y ’ 2x Hypotenuse ’ 2 multiplied by shorter leg.
[pic] Substitute [pic] for x.
[pic] Simplify.
Find the values of x and y. Give your answers in simplest radical form.
5. [pic] 6. [pic]
7. [pic] 8. [pic]
Trigonometric Ratios
You can use special right triangles to write trigonometric ratios as fractions.
[pic]
[pic]
[pic]
So sin 45° ’ [pic].w
Write each trigonometric ratio as a fraction and as a decimal
rounded to the nearest hundredth.
1. sin K 2. cos H
3. cos K 4. tan H
Use a special right triangle to write each trigonometric ratio as a fraction.
5. cos 45° 6. tan 45°
7. sin 60° 8. tan 30°
Trigonometric Ratios continued
You can use a calculator to find the value of trigonometric ratios.
cos 38° ≈ 0.7880107536 or about 0.79
You can use trigonometric ratios to find side lengths of triangles.
Find WY.
[pic] Write a trigonometric ratio that involves WY.
[pic] Substitute the given values.
[pic] Solve for WY.
WY ≈ 9.65 cm Simplify the expression.
Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.
9. sin 42° 10. cos 89°
11. tan 55° 12. sin 6°
Find each length. Round to the nearest hundredth.
13. DE 14. FH
[pic] [pic]
15. JK 16. US
[pic] [pic]
Solving Right Triangles
Use the trigonometric ratio sin A ( 0.8 to determine which angle of the triangle
is ∠A.
[pic] [pic]
[pic] [pic]
’ 0.6 ’ 0.8
Since sin A ’ sin ∠2, ∠2 is ∠A.
If you know the sine, cosine, or tangent of an acute angle measure, then you can
use your calculator to find the measure of the angle.
Use the given trigonometric ratio to determine which angle of the triangle
is ∠A.
[pic]
1. [pic] 2. [pic]
3. cos A ’ 0.5 4. [pic]
Use your calculator to find each angle measure to the nearest degree.
5. sin−1 (0.8) 6. cos−1 (0.19)
7. tan−1 (3.4) 8. [pic]
Angles of Elevation and Depression
[pic]
Classify each angle as an angle of elevation or an angle of depression.
1. ∠1 2. ∠2
[pic] [pic]
Use the figure for Exercises 3 and 4. Classify each angle as an angle of elevation or an angle of depression.
3. ∠3
4. ∠4
Use the figure for Exercises 5–8. Classify each angle as an angle of elevation
or an angle of depression.
5. ∠1
6. ∠2
7. ∠3
8. ∠4
Angles of Elevation and Depression continued
You can solve problems by using angles of elevation and angles of depression.
Sarah is watching a parade from a 20-foot balcony. The angle of depression
to the parade is 47°. What is the distance between Sarah and the parade?
Draw a sketch to represent the given information. Let A represent
Sarah and let B represent the parade. Let x represent the distance
between Sarah and the parade.
m∠B ’ 47° by the Alternate Interior Angles Theorem. Write a sine ratio
using ∠B.
[pic]
x sin 47° ’ 20 ft Multiply both sides by x.
[pic] Divide both sides by sin 47°.
27 ft ≈ x Simplify the expression.
The distance between Sarah and the parade is about 27 feet.
9. When the angle of elevation to the sun 10. A person snorkeling sees a turtle on the
is 52°, a tree casts a shadow that is ocean floor at an angle of depression of
9 meters long. What is the height of 38°. She is 14 feet above the ocean floor.
the tree? Round to the nearest tenth How far from the turtle is she? Round to
of a meter. the nearest foot.
[pic] [pic]
11. Jared is standing 12 feet from a 12. Maria is looking out a 17-foot-high
rock-climbing wall. When he looks up window and sees two deer. The angle of
to see his friend ascend the wall, the depression to the deer is 26°. What is the
angle of elevation is 56°. How high up horizontal distance from Maria to the
the wall is his friend? Round to the deer? Round to the nearest foot.
nearest foot.
-----------------------
Take the positive square root and simplify.
|Pythagorean Triples |Not Pythagorean Triples |
|3, 4, 5, |2, 3, 4 |
|5, 12, 13 |6, 9, [pic] |
|Pythagorean Inequalities Theorem |
|[pic] | [pic] |
|If c2 > a2 + b2, then nABC is obtuse. |If c2 < a2 + b2, then nABC is acute. |
m∠C < 90o
m∠C ( 90o
|Theorem |Example |
|450-450-900 Triangle Theorem |[pic] |
|In a 450-450-900 triangle, both legs are congruent and the length of| |
|the hypotenuse is [pic] times the length of a leg. | |
|Theorem |Examples |
|300-600-900 Triangle Theorem |[pic] |
|In a 300-600-900 triangle, the length of the hypotenuse is 2 | |
|multiplied by the length of the shorter leg, and the longer leg | |
|is [pic] multiplied by the length of the shorter leg. | |
|Trigonometric Ratios |
|[pic] |
|[pic] |
|[pic] |
hypotenuse
leg opposite ∠A
leg adjacent to ∠A
|Inverse Trigonometric Functions |
|Symbols |Examples |
|sin A ’ x ⇒ sin−1 x ’ m∠A |[pic] |
|cos B ’ x ⇒ cos−1 x ’ m∠B |[pic] |
|tan C ’ x ⇒ tan−1 x ’ m∠C |tan 76° ≈ 4.01 ⇒ tan−1 (4.01) ≈ 76° |
An angle of depression is formed by a horizontal line and a line of sight below it.
An angle of elevation is formed by a horizontal line and a line of sight above it.
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