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Wize High School Geometry Textbook (Common Core) > Trigonometry of Right Triangles

Primary Trig Ratios (sin⁡\sin, cos⁡\cos, tan⁡\tan)

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How to Label a Right-Angle Triangle?

Naming a Triangle

When labelling a triangle, we usually name the angles with
CAPITAL letters
and the sides with
lower case letters
.

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The 3 Sides in a Right-Angle Triangle

  1. the hypotenuse--
    longest side of the triangle, it is across from the right angle
  2. the opposide side --
    side opposite from the angle of interest/given angle
  3. the adjacent side --
    remaining side next to the angle of interest/given angle
Watch Out!
The opposide and adjacent sides CHANGE, depending on which angle we are focusing on.


Practice: Labelling a Right-Angle Triangle

Label the sides of the following triangle with O (for opposite), A (for adjacent), and H (for hypotenuse) according to the given angle.


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Primary Trigonometric Ratios--Sin, Cos, Tan

sin⁡θ=OppHyp          cos⁡θ=AdjHyp          tan⁡θ=OppAdj\large \boxed{\sin\theta=\dfrac{Opp}{Hyp}~~~~~~~~~~ \cos\theta=\dfrac{Adj}{Hyp}~~~~~~~~~~ \tan\theta=\dfrac{Opp}{Adj}}

How do we remember this?

SOHsinθ=Opposite/Hypotenuse\begin{array}{ccccc} \Large{\text{S}}&&\Large{\text{O}}&&\Large{\text{H}}\\ \text{\colorTwo{s}in}\theta&\scriptsize{=}&\text{\colorTwo{O}pposite}&\scriptsize{/}&\text{\colorTwo{H}ypotenuse} \end{array}

CAHcosθ=Adjacent/Hypotenuse\begin{array}{ccccc} \Large{\text{C}}&&\Large{\text{A}}&&\Large{\text{H}}\\ \text{\colorTwo{c}os}\theta&\scriptsize{=}&\text{\colorTwo{A}djacent}&\scriptsize{/}&\text{\colorTwo{H}ypotenuse} \end{array}

TOAtanθ=Opposite/Adjacent\begin{array}{ccccc} \Large{\text{T}}&&\Large{\text{O}}&&\Large{\text{A}}\\ \text{\colorTwo{t}an}\theta&\scriptsize{=}&\text{\colorTwo{O}pposite}&\scriptsize{/}&\text{\colorTwo{A}djacent} \end{array}


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Calculator Time!

Watch Out!
*Make sure your calculator is in the correct degree (D) mode.

Use your calculator to evaluate the following trig ratios:
1.) sin⁡30°=\sin30\degree=
0.5

2.) cos⁡ 50°=\cos\ 50\degree=
0.6428

3.) tan⁡23.5°=\tan23.5\degree=
0.4348

4.) sin⁡211°=\sin 211\degree=
-0.5150

5.) cos⁡180°=\cos180\degree=
-1

6.) tan⁡90°=\tan90\degree=
error (undefined)

Practice: Primary Trig Ratios

Use a calculate to evaluate the following.

a) cos⁡60°=\cos60\degree=

b) sin⁡200°=\sin200\degree=

c) tan⁡190°=\tan190\degree=

d) sin⁡300°=\sin300\degree=

e) cos⁡0°=\cos0\degree=

f) tan⁡270°=\tan270\degree=

*Round your answer to 2 decimal places. Enter DNE if the answer is undefined (error).
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Practice: Primary Trig Ratios

Given the following triangle, find all of the primary trig ratios.


sin⁡θ=OppHypsin⁡θ=1213cos⁡θ=AdjHypcos⁡θ=513tan⁡θ=OppAdjtan⁡θ=125\begin{array}{ccccc} \begin{array}{rcl} \sin\theta&=&\dfrac{Opp}{Hyp}\\[1em] \sin\theta&=&\dfrac{12}{13} \end{array} && \begin{array}{rcl} \cos\theta&=&\dfrac{Adj}{Hyp}\\[1em] \cos\theta&=&\dfrac{5}{13} \end{array} && \begin{array}{rcl} \tan\theta&=&\dfrac{Opp}{Adj}\\[1em] \tan\theta&=&\dfrac{12}{5} \end{array} \end{array}

Practice: Primary Trig Ratios

Given the following triangle, find all of the primary trig ratios.

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Example: Solving Equations w/ Primary Trig Ratios

Solve the following equations for xx.

a) tan⁡30°=x10\tan30\degree=\dfrac{x}{10}
tan⁡30°=x10tan⁡30°×10=x10×10tan⁡30°×10=x(0.577)×10≈x5.77≈xx≈5.77\begin{array}{rcl} \tan30\degree&=&\dfrac{x}{10}\\[1em] \tan30\degree\scriptsize{\colorTwo{\times10}}&=&\dfrac{x}{10}\scriptsize{\colorTwo{\times10}}\\[1em] \tan30\degree\times 10&=&x\\[1em] (0.577)\times10&\approx&x\\[1em] 5.77&\approx&x\\[1em] x&\approx&5.77 \end{array}

Therefore, the exact answer for xx is 10×tan⁡30°10\times\tan30\degree, which is approximately 5.775.77.
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b) sin⁡45°=3x\sin45\degree=\dfrac{3}{x}
sin⁡45°=3xsin⁡45°×x=3x×xsin⁡45°×x=3sin⁡45°×xsin⁡45°=3sin⁡45°x=3sin⁡45°x≈=30.707x≈4.24\begin{array}{rcl} \sin45\degree&=&\dfrac{3}{x}\\[1em] \sin45\degree\scriptsize\colorTwo{\times x}&=&\dfrac{3}{x}\scriptsize\colorTwo{\times x}\\[1em]\\ \sin45\degree \times x&=&3\\[1em] \dfrac{\sin45\degree \times x}{\scriptsize\colorTwo{ \sin45\degree}}&=&\dfrac{3}{\scriptsize\colorTwo{\sin45\degree}}\\[1em] x&=&\dfrac{3}{\sin45\degree}\\[1em] x&\approx&=\dfrac{3}{0.707}\\[1em] x&\approx&4.24 \end{array}

Therefore, the exact answer is 3sin⁡45°\dfrac{3}{\sin 45\degree}, which is approximately 4.24.

Practice: Solving Equations w/ Primary Trig Ratios

Solve for xx in the equation cos⁡135°=x8\cos135\degree=\dfrac{x}{8}

Practice: Solving Equations w/ Primary Trig Ratios

Solve for xx in the equation sin⁡150°=−5x\sin150\degree=-\dfrac{5}{x}

Practice: Primary Trig Ratios

A ramp to a house has an angle that measures 30°30\degree as shown in the picture below.

Using our calculators, we can see that sin⁡30°=12\sin30\degree=\dfrac{1}{2}.

True or False?
The height of the ramp must measure 1 meter and the length must be 2 meters. Justify your answer.