Transcript Extra Practice
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By: Marc Hensley
Right Triangles and Trigonometry
The Pythagorean Theorem is a relation in Geometry between the 3 sides of a right triangle The theorem states: In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs
Generally speaking, the formula is written as: a 2 + b 2 = c 2
where a and b are the legs, and c is the hypotenuse.
c a
b
7 x x 5 13 24 7
2
+ 24
2
= x
2
49 + 576 = x
2
625 = x
2
25 = x 5
2
+ x
2
= 13
2
25 + x
2
= 169 x
2
= 144 x = 12
Solve for x.
a) 38 c) 40
16 34 x
b) 30
d) 45
16 2 + x 2 = 34 2
256 + x
2
= 1156 x
2
= 900 16 x 34
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x = 30
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A special right triangle is a right triangle whose sides are in a particular ratio.
Recognizing special right triangles in geometry can help you to answer some questions quicker.
Types of special triangles
There are 2 main types of special right triangles: 1) The 45-45-90 2) The 30-60-90
Click on the triangle you want to learn about.
The lengths of the sides of a 45°- 45°- 90° triangle are in the ratio of 1 : 1 : √2
x
x
The lengths of the sides of a 30°- 60°- 90° triangle are in x
x√3
Solve for the missing parts of the triangle. Round your answers to the nearest tenth.
7.4
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z
x
Solve for the missing parts of the triangle. Round your answers to the nearest tenth.
x
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y
x = 7.4
y = 10.5
z = 45
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x = 6.5
y = 11.3
z = 60
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Right triangles have 3 special formulas that ONLY WORK in right triangles.
They are: sin = Opposite hypotenuse A c b cos = adjacent hypotenuse tan = opposite adjacent C a B
Draw a picture depicting the situation.
Be sure to place the degrees INSIDE the triangle.
Place a stick figure at the angle as a point of reference.
Thinking of yourself as the stick figure, label the opposite side (the side across from you), the hypotenuse (across from the right angle), and the adjacent side (the leftover side).
Figure out which pair of sides the problem deals with (for example: opposite and hypotenuse) and choose the correct equation (in our example, sin)
In right triangle ABC, hypotenuse AB=15 and angle A=35º.
Find leg BC to the nearest tenth.
1) First, draw a picture, and label everything you know.
2) Then, figure out which trig function we will use. In this case, we will use sin.
3) Set up the equation.
4) Solve for x.
sin 35 = 15 x = 15 sin 35 x = 8.6
X
In triangle RST, angle R is a right angle, angle S has measure of 65 degrees, and RS = 9. Find the measure of ST.
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cos 65 = 9 x x * cos 65 = 9 x = 9 cos 65 x = 21. 3 9 S 65 R 90 x
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T
We use the LAW OF SINES to solve triangles that are not right triangle.
The law of sines states the following: The sides of a triangle are to one another in the same ratio as the sines of their opposite angles.
What does that mean?
We can use this triangle to set up the equation…..
B c sin A = a sin B b = sin C c A b
a C
The three angles of a triangle are 40°, 75°, and 65°. When the side opposite the 75° angle is 10 cm, how long is the side opposite the 40° angle?
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sin 75 10 = sin 40 x x = 6.7
The three angles of a triangle are A = 30°, B = 70°, and C = 80°. If side a = 5 cm, find sides b and c.
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B = 9.4
C = 9.9
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Angles of elevation and depression are angles that are formed with the horizontal. If the line of sight is upward from the horizontal, the angle is an angle of elevation; if the line of sight is downward from the horizontal, the angle is an angle of depression.
Using these types of angles and some
between points.
indirectly calculate heights of objects or distances Alternatively, if the heights or distances are known, the angles can be determined.
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Suppose a flagpole casts a shadow of 20 feet. The angle of elevation from the end of the shadow to the top of the flagpole measures 50°. Find the height of the flagpole.
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tan 50 = x 20 x = 23.8
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Suppose a tree 50 feet in height casts a shadow of length 60 feet. What is the angle of elevation from the end of the shadow to the top of the tree with respect to the ground?
First, we draw the picture Then, set up the equation: tan x = 50 60 • Finally, solve for x.
x = tan -1 ( ) = 39.8
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The Law of Cosines (also known as the Cosine Rule or
Cosine Law) is a generalization of the Pythagorean Theorem
Basically, the Pythagorean Theorem requires there to be a right angle in a triangle But, if there is not, the Law of Cosines can be used
For a triangle with sides a, b, and c opposite (respectively) the angles A, B, and C, the Law of Cosines states: c 2 = a 2 + b 2 - 2ab·cos(C) a 2 = b 2 + c 2 – 2bc·cos(A) b 2 = a 2 + c 2 - 2ac·cos(B)
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The law of cosines should be used when use have SSS or SAS in a triangle, like the one picture here… The law of cosines states: s 2 = r 2 + t 2 – 2rtcos(S), or r 2 = s 2 + t 2 - 2stcos(R), or t 2 = r 2 + s 2 – 2rscos(T)
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In triangle ABC, you are given a = 10, B = 32 o Find the measure of side b.
and c = 15. First, write out the equation: b 2 = a 2 + c 2 -2 ac cos B.
So, b 2 = 100 + 225 – 2*(10)*(15)*cos 32 º
b
2 = 325 - 300 (0.848048096)
b
2 = 325 - 254.4 b 2 = 70.59. Therefore, b = 8.4.
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In triangle ABC, you are given A = 28 o , b = 14, c = 10. Solve for side a.
First, write out the equation: a 2 = b 2 + c 2 - 2bc*cos A.
So, a 2 = 296 + 100 – 2*(14)*(10)*cos 28 º
a
2 = 396 - 280 (0.8829475929)
a
2 = 396 – 247.2
a 2 = 148.8. Therefore, a = 12.2.
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