Transcript Naming the Trigonometric Ratios

Naming and Using the Trigonometric Ratios
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The trigonometric ratios which have been explored in 5.4
(i.e. opp/hyp, adj/hyp opp/adj) are given names as
follows…
The ratio opp/hyp is known as sine. Sine = opp/hyp
The ratio adj/hyp is known as cosine. Cosine = adj/hyp
The ratio opp/adj is known as tangent. Tangent = opp/adj
That allows us to change our language slightly. For
example, instead of saying something like the opp/hyp
ratio for 40o is 0.64, we can simply say the sine of 40o is
0.64
These names and subsequent formulas will play a role in
our work when finding missing sides and angles.
Naming and Using the Trigonometric Ratios
Of course what is most important about the ratios
sine, cosine and tangent is, for any given reference
angle, they are constants.
 Ex.
40o
Even though the above triangles are different sizes;
the ratio opp/hyp (sine) is the same in both (0.64)
the ratio adj/hyp (cosine) is the same in both(0.77)
the ratio opp/adj (tangent) is the same in both(0.84)
It is because they are constants we can determine
missing sides and angles.
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Naming and Using the Trigonometric Ratios
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Lets try using the new names/formulas and finding a missing side and a
missing angle.
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Missing side
600
?
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20
Since the problem involves the adj. and hyp. sides we know we will be using
the ratio involving those sides which we now know is called cosine so start
the problem with that formula cos = adj/hyp
The cos of 600 is always the same (0.5) so in our triangle the adj (?) divided
by the hyp (20) must equal the cos 600 so….
cos = adj/hyp
cos 600 = ?/20
Since the cos 600 is 0.5 lets change it to that
0.5 = ? /20
Solve the equation
0.5 x 20 = ? /20 x 20
10 = ?
Naming and Using the Trigonometric Ratios
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Missing Angle
252
?
300
The problem involves the sides opp. and adj. so we know we will be using the
ratio involving those sides which we know is called tangent so start the
problem with that formula tan = opp/adj
Since we know the sides we can find the ratio and then will just need to
determine when tangent equals that ratio so …
tan = opp/adj
tan ? = 252/300
We are used to the ratios in decimal form so change it
tan ? = 0.84
For what angle does tan = 0.84? Check your table or book.
? = 400
Naming and Using the Trigonometric Ratios
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One more missing side problem.
60
?
300
Since the problem involves the opp. and hyp. sides we know we will be
using the ratio involving those sides which we now know is called sine so
start the problem with that formula sin = opp/hyp
The sin of 300 is always the same (0.5) so in our triangle the opp (60)
divided by the hyp (?) must equal the sin 300 so….
sin = opp/hyp
sin 300 = 60/?
Since the sin 300 is 0.5 lets change it to that
0.5 = 60/?
Solve the equation. This equation may seem a little more difficult to solve
due to the location of the variable. It may help to rearrange it. Think about
this. If 6/2 = 3, what other arithmetic fact do you know?
2x3=6
Apply that to our equation. If 60/? = 0.5 then
0.5 x ? = 60
It should be clear what to do now to solve.
0.5 x ? = 60
0.5
0.5
? = 120