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Sine, Cosine, Tangent Ratios

Practice writing the ratios

Video Tutorial- How to write sohcahtoa ratios, given side lengths

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What do we mean by 'ratio' of sides?

Diagram of Sine Ratios Sine, cosine and tangent of an angle represent the ratios that are always true for given angles. Remember these ratios only apply to right triangles.

The 3 triangles pictured on the left illustrate this.

Although the side lengths of the right triangles are different, each one has a 37° angle, and as you can see, the sine of 37 is always the same! In other words, sin(37) is always .6 !
(Note: I rounded to the nearest tenth) That's, of course, why we can use a calculator to find these sine, cosine and tangent ratios.
Practice Problems



Want to try some word problems on this topcic? Click here
problemIn the triangle on the left, what side is adjacent to $$\angle MLN$$ ?
Adjacent Side


problem Calculate $$cos(\angle MLN)$$
Cosine


problem Calculate $$cos(\angle MNL)$$
Cosine


problem What is the sine ratio of $$\angle ACB$$ in the triangle on the right?

Answer
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pick angle problem Which angle on the left has a tangent of $$\frac{3}{4}$$ ?
Answer
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challenge Problem Challenge Problem
Be careful!
pick angle In the triangle on the left, which angle has a sine ratio of 2.6?
Answer
Word Problems



problemIn $$\triangle JKL$$, sin(k) = $$\frac{3}{5} $$, what is tan(k)?
Step 1
It's easiest to do a word problem like this one, by first drawing the triangle and labelling the sides. We know the opposite side of $$ \angle K$$ and we know the hypotenuse
sine picture

To get the tangent ratio we need to know the length of the adjacent side
How can we find the length of the adjacent side?
Adjacent Side
Use the pythagorean theorem !
$$ a^2 + b^2 = c^2 \\ 3^2 + b^2 = 5^2 \\ b^2 = 5^2- 3^2 = 25-9 = 16 \\ b = 4 $$

Now, use the tangent ratio!
Last Step
$$ tan( k ) = \frac{opposite}{adjacent} \\ tan( k ) = \frac{3}{4} $$
sine picture 3


problem In $$\triangle ABC , cos(b) = \frac{7}{25} $$, what is sin(b)?
Step 1
Draw this triangle and label the sides:
cosine word problem Remember that the cosine ratio =$$\frac{adjacent}{hypotenuse}$$.
How can we find the length of the opposite side? (Remember that sine involves the opposite so we need to find that somehow)
Opposite Side
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