Download Right Triangle Trigonometry

Right Triangle Trigonometry
Six New Functions
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Consider a generic right triangle.
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We are going to define six new functions that help measure the shape of this triangle.
These new functions are called trigonometric functions, or trig functions for short.
If you consider the situation carefully, you might realize that the shape (not size) of
the above right triangle is completely determined by the angle θ. So, when we define
our trig functions,the independent variable (or the domain values) will be angle
measures. The dependent variable (or the range values) will be ratios of side
lengths.
Remark About notation...
In any right triangle, the side not touching the right angle is called the
hypotenuse or hyp. In a right triangle with one acute angle labeled
θ (as above), the side not touching θ is called the opposite side or
opp and the side touching θ that is not the hypotenuse is called the
adjacent side or adj.
Note these labels change depending on which angle we label θ!
Def. The six trigonometric functions of the angle θ are
defined as:
Definition
Read as:
Definition
Read as:
sin θ =
opp Sine of theta equals
hyp opposite over hypotenuse
csc θ =
hyp Cosecant of theta equals
opp hypotenuse over opposite.
cos θ =
adj Cosine of theta equals
hyp adjacent over hypotenuse
sec θ =
hyp Secant of theta equals
adj hypotenuse over adjacent.
tan θ =
opp Tangent of theta equals
adj opposite over adjacent
cot θ =
adj Cotangent of theta equals
opp adjacent over opposite.
Remark Some people use the acronym SOHCAHTOA to remember the definitions of sin, cos, and tan.
SOH → Sin is Opp over Hyp
CAH → Cos is Adj over Hyp
TOA → Tan is Opp over Adj
Example For the triangle below, find the values of the six trig functions of θ.
Example For the triangle below, find the values of the six trig functions of θ.
Three Special Angles: 30◦ = π6 , 45◦ = π4 , and 60◦ =
π
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From the above examples we begin to understand that given a right triangle, evaluating
trig functions is pretty straightforward. Which means, for example, to find sin 45◦ , we
first draw a right triangle with an acute angle of 45◦ . Or, to find cos π3 , we first draw
a right triangle with an acute angle of π3 . It turns out that there are only three acute
angles for which we can easily draw such triangles.
Example Find sin 45◦ and cos 45◦ .
Example Find sin 30◦ and cos 30◦ .
Remark Notice in the last example that if we rename θ to be the 60◦ angle, and
then rename our opp and adj sides accordingly, we can then calculate
sin 60◦ and cos 60◦ . See the diagram below.
Remark You must memorize the values of the six trig functions of 30◦ (or π6
radians), 45◦ (or π4 radians), and 60◦ (or π3 radians). To do this, I
suggest memorizing the two triangles we derived in the above examples.
Draw them below!
Cofunction Identities
Notice, using our special traingles, that sin 30◦ = cos 60◦ or tan 60◦ = cot 30◦ this
principle generalizes to all complementary angles and all pairs of “co” functions.
Formula For 0◦ < θ < 90◦ we have:
sin θ = cos(90◦ − θ)
cos θ = sin(90◦ − θ)
sec θ = csc(90◦ − θ)
csc θ = sec(90◦ − θ)
tan θ = cot(90◦ − θ)
cot θ = tan(90◦ − θ)
If θ is in radians, replace 90◦ with
π
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Applications
In many applications you will have to solve for one side length in a right triangle given
only an angle and one other side. Consider the following example.
Example For each triangle below, solve for x. First write your answer in terms
of trig functions, and then approximate using a calculator.
a
b