Trigonometry Lesson-1 Definitions & Right Triangles |
The Trigonometric Functions :
Let's analyze the word "Trigonometry". The prefix tri means 3 -- as in trimester, triangle or tricycle. The last part of the word nometry -- is a Latin term for numbering or measuring.
So Trigonometry deals with the measures of 3 sided figures -- or triangles.
This form of math was first used around 160 B.C. by Hipparchus, an ancient Greek fellow interested in astronomy and land measuring. We still use it today to measure distances on the ground for surveying, measuring the height of a building, and especially for navigation.
Trigonometry defines three functions and their reciprocals based on the ratio of the lengths of specific sides of a right triangle. The ancients noticed that -- no matter how long or short the sides of any right triangle, the ratio of the lengths of the 3 sides, taken in pairs, remained constant, so they assigned specific names to the three ratios or functions -- sine, cosine, and tangent.
Since they did, we've been using these functions to assist us in every known science -- because math is the language of science -- all forms of science -- chemistry, physics, biology -- are ruled by the natural mathematical relations which exist in our world and Trigonometry helps us understand them.Without trig, land surveyors, navigators and astronomers would be lost and frustrated.
One final request before we get started. Please don't refer to the sine function as "sin" -- the word we hear in church. "Sin" is an abbreviation of sine, so it is pronounced "sign". Sin is for Philosophy and Religion courses. Sine is for math.
Definitions
There are really 6 trig functions defined on angles, however, 3 of them are the reciprocals of the others, and our calculators list only three. Makes sense -- if we need a reciprocal function,
we hit the x^{ -1} or 1/x button and we've got it -- so why waste buttons for no good reason?
The three trig functions on our calculators are:
sine (sin) | cosine (cos) | tangent (tan) |
Each trig function represents a RATIO of 2 sides in a right triangle.
The sine, cosine, and tangent of an angle are CONSTANT.
This means that sin 30º is always 0.5 regardless of the lengths of the sides of the triangle.
Note: the side opposite the 90° angle is always called the hypotenuse.
As we can see in the diagram, the sides opposite the angles are labeled with the same letters but in lower case. The side opposite angle A is labeled " a " etc.
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Definition: the angle of elevation to an object is measured from the horizontal upwards.
Definition: the angle of depression to an object is measured from the horizontal downwards.
SOLVING RIGHT TRIANGLES
When we solve a right triangle, we will be given 2 pieces of information and asked to find the remaining angles and sides. Since the acute angles of a right triangle are complementary, (they add to 90° ) , if we're given one of the acute angles, we can find the second by subtracting the first from 90°. The sides are found using trig functions.
Example 1:
Solve the right triangle if angle A = 35° and b = 12 cm . Angle C = 90° .
When we're given the measures of 2 sides, we need to find the 3rd side and the 2 acute angles. We use the inverse trig functions shown as sin^{ - 1}, cos^{ - 1}, tan^{ - 1} on our calculators to do this. (on most calculators they are "second" or "shift" functions. They're often found over the trig functions.
The sine of an angle tells us the ratio of the opposite side to the hypotenuse. The inverse sine (arcsin or sin^{ - 1} ) of a ratio tells us the ANGLE with sine equal to that ratio.
We know from the 30° , 60° , 90° triangle that sin 30° = ½. So arcsin ½ = 30°. If the answers are to be given in degrees, make sure your calculator isn't set to radians in the MODE menu.
Say we know that sin A = 0.7 and we need to find angle A . We enter sin^{ - 1} 0.7 and the calculator will display the measure of angle A. It is 44.43°.
The function sin^{ - 1} is also called arcsin. This is the best notation. See note below.
The other inverse trig functions are arccos and arctan.
Important Note:
Beware!!! In Algebra, x^{ -1} means the reciprocal of x.
In the notation for inverse trig functions, the -1 exponent does not mean reciprocal -- it indicates the inverse function. The reciprocal of the sine function is the cosecant function not the arcsin function. Because this causes confusion, it is best to always use arcsin, arccos, arctan, etc. to indicate an inverse trig function.
With y = sin x, we plug in angle values for x and generate ratio values for y.
When we have y = arcsin x, we substitute ratio values for x and generate angle values for y.
So, if x = 45^{ }°, sin x = 0.7071 which means that arcsin 0.7071^{ } = 45°
We substitute an angle value for x in y = sin x; we get a ratio.
We substitute a ratio value for x in y = arcsin x; we get an angle.
So the statement y = sin x is identical to x = arcsin y since x is the angle whose sine = y.
Note:
We never write the trig function names without an argument --
something to represent the angle to which we apply the trig function.
In the statement y = sin x, x is called the argument.
In the statement y = sin 37° , 37° is the argument.
So, don't write sin = 0.4571 and lose half a mark, write sin A = 0.4571 to get full credit. Form is extremely important in the language called math! Don't mess with it. It's been around longer than you -- so learn it, apply it -- and don't make a fuss.
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Example 2
Solve the right triangle ABC in the diagram:
Since we know the opposite and the adjacent sides of angle A, we will solve for it first
using the tangent function. We know that:
tan A = = 0.52518 therefore angle A = arctan (0.52518) = 27.71°.
therefore angle B = 90^{ }° - 27.71° = 62.29° .
To find c , we could use the Pythagorean theorem but let's use a trig function.
Now that we know angle A = 27.71° , we can use sin A to find c .
sin 27.71°= , so c = = 15.7 cm.
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Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.
Practice :
1) Solve these right triangles. (note: the right angle is always at C)
a) a = 4 , angle A = 35° | b) b = 5, angle A = 27° | c) a = 13, b= 9 |
d) a = 101, b = 116 | e) a = 12.5 , b = 8.7 | f) angle B = 6.2°, c = 3720 |
( solutions: a, b, c, d, e, f )
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2) A rectangle is 26 cm by 14 cm.
( solutions )
3) A 20m ladder is placed against a wall so that the foot of the ladder is 5m from the wall.
( solutions )
4) The lengths of the shadows of two vertical poles are 72.5 cm and 40.3 cm respectively.
The first pole is 25 cm taller than the second pole.
( solutions )
5) A lighthouse 25 m. high stands on the top of a cliff.
From a point on the seashore, the angles of elevation to the top and bottom of the lighthouse are 47.2° and 45.22° respectively. Find the height of the cliff.
( solutions )
6) From the top of a building and from a window 30 m. below the top, the angles of depression to a car on the street are 15.67° and 10° respectively.
( solutions )
7) From a third story window, the angle of depression of the foot of a buiding across
the street is 37°. The angle of elevation to the top of the building is 51°. If the street is 25 m. wide, find the height of the building.
( solutions )
8) A particle with an acceleration of 1.9 m/sec^{2} makes an angle of 22.5° with the x-axis.
Find the x and y components of the acceleration.
( solutions )
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Solutions
1) Solve these right triangles. (note: the right angle is always at C)
a)
angle B = 90° - 35° = 55°, tan 55° = b = 4 tan 55° = 5.71
sin 35° =
angle B = 90° - 27° = 63°, tan 27° = a = 5 tan 27° = 2.55
cos 27° = = 5.61
tan A = so, angle A = arctan 1.44... = 55.3°
angle B = 90° - 55.3° = 34.7°
sin 55.3° =
tan A = , therefore angle A = arctan 0.87069 = 41.05°
angle B = 90° - 41.05° = 48.95°
sin 41.05° =
tan B = , so angle B = arctan 0.696 = 34.84°
angle A = 90° - 34.84° = 55.16°
cos 55.16° =
angle A = 90° - 6.2° = 83.8°.
sin 6.2° = b = 3720 sin 6.2° = 401.76
cos 6.2° = a = 3720 cos 6.2°= 3698.24
a) Tan DAC = , so angle DAC = arctan 0.538462 = 28.3°
b) d² = 26² + 14² so d = = 29.53 cm.
3) A 20m ladder is placed against a wall so that the foot of the ladder is 5m from the wall.
a) What angle does the ladder make with the wall?
b) How high up the wall does the ladder reach?
a) sin B = so angle B = arcsin 0.25 = 14.48°
b) tan 14.48° =
4) The lengths of the shadows of two vertical poles are 72.5 cm and 40.3 cm respectively.
The first pole is 25 cm taller than the second pole.
a) Since triangles CED and CBA are similar, we can solve for y:
72.5y = 40.3y + 1007.5
so 32.2y = 1007.5, and y = 31.29
We're looking for either angle D or angle A since they're both the angle of elevation of the sun.
tan D = = 0.776427 so angle D= arctan 0.776427 = 37.83°
b) We've already found the shorter pole when we found y, so, the shorter pole is 31.29 cm tall and the longer pole is 31.29 + 25 = 56.29 cm.
In this question, we have to solve for an unknown in 2 different ways and then equate the two expressions in order to solve for one of the missing parts.
We find 2 expressions for a and equate them.
so, since a = a, we set the 2 expressions equal to each other, then solve for y.
y(tan 47.2°) = (y + 25)(tan 45.22°)
when we multiply out the brackets and collect like terms we get:
y(tan 45.22° – tan 47.2°) = – 25(tan 45.22°) = – 25.192727
so, , so the cliff is 348.97 m high.
Again, we'll find 2 expressions for a and equate them:
therefore (y + 30)(tan 10°) = y ( tan 15.67°)
so
The answer for b) is the building is 50.77 + 30 = 80.77 m.
since , and we know that y = 50.77,
= 287.92 m.
From the diagram it is obvious that the height of the building is
the sum of DE and EC (y + x) and AE = 25 m.
tan 51° = y = 25 tan 51° = 30.87 m
tan 37° = x = 25 tan 37° = 18.84 m
x + y = 30.87 m + 18.84 m = 49.71 m.
cos 22.5° = x = 1.9 cos 22.5° = 1.76 m/sec^{2}
sin 22.5°= y = 1.9 sin 22.5°= 0.73 m/sec^{2}