Wize High School Grade 10 Math Textbook > Trigonometry of Right Triangles

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

How to Label a Right-Angle Triangle?

Naming a Triangle

When labelling a triangle, we usually name the angles with ________ and the sides with ________.

Image illustrating How to Label a Right-Angle Triangle?

The 3 Sides in a Right-Angle Triangle

the hypotenuse--________
the opposide side -- ________
the adjacent side -- ________

Primary Trigonometric Ratios--Sin, Cos, Tan

$Image illustrating Primary Trig Ratios--$\sin$, $\cos$, $\tan$$

$\large \boxed{\sin\theta=\dfrac{Opp}{Hyp}~~~~~~~~~~ \cos\theta=\dfrac{Adj}{Hyp}~~~~~~~~~~ \tan\theta=\dfrac{Opp}{Adj}}$

How do we remember this?

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

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

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

Calculator Time!

Use your calculator to evaluate the following trig ratios:

1.) $\sin30\degree=$ ________

2.) $\cos\ 50\degree=$ ________

3.) $\tan23.5\degree=$ ________

4.) $\sin 211\degree=$ ________

5.) $\cos180\degree=$ ________

6.) $\tan90\degree=$ ________

Practice: Primary Trig Ratios

Use a calculate to evaluate the following.

a) $\cos60\degree=$

b) $\sin200\degree=$

c) $\tan190\degree=$

d) $\sin300\degree=$

e) $\cos0\degree=$

f) $\tan270\degree=$

*Round your answer to 2 decimal places. Enter DNE if the answer is undefined (error).

a)b)c)d)e)f)

I don't know

Practice: Primary Trig Ratios

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

$\begin{array}{ccccc} \begin{array}{rcl} \sin\theta&=&\dfrac{Opp}{Hyp}\\[1em] \sin\theta&=&\dfrac{12}{13} \end{array} && \begin{array}{rcl} \cos\theta&=&\dfrac{Adj}{Hyp}\\[1em] \cos\theta&=&\dfrac{5}{13} \end{array} && \begin{array}{rcl} \tan\theta&=&\dfrac{Opp}{Adj}\\[1em] \tan\theta&=&\dfrac{12}{5} \end{array} \end{array}$

Practice: Primary Trig Ratios

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

$\Large{\color{black}{\sin\theta=}}$$\Large{\color{black}{\cos\phi=}}$$\Large{\color{black}{\tan\phi=}}$$\Large{\color{black}{\tan\theta=}}$

I don't know

Example: Solving Equations w/ Primary Trig Ratios

Solve the following equations for $x$ .

a) $\tan30\degree=\dfrac{x}{10}$

$\begin{array}{rcl} \tan30\degree&=&\dfrac{x}{10}\\[1em] \tan30\degree\scriptsize{\colorTwo{\times10}}&=&\dfrac{x}{10}\scriptsize{\colorTwo{\times10}}\\[1em] \tan30\degree\times 10&=&x\\[1em] (0.577)\times10&\approx&x\\[1em] 5.77&\approx&x\\[1em] x&\approx&5.77 \end{array}$

Therefore, the exact answer for $x$ is $10\times\tan30\degree$ , which is approximately $5.77$ .

b) $\sin45\degree=\dfrac{3}{x}$

$\begin{array}{rcl} \sin45\degree&=&\dfrac{3}{x}\\[1em] \sin45\degree\scriptsize\colorTwo{\times x}&=&\dfrac{3}{x}\scriptsize\colorTwo{\times x}\\[1em]\\ \sin45\degree \times x&=&3\\[1em] \dfrac{\sin45\degree \times x}{\scriptsize\colorTwo{ \sin45\degree}}&=&\dfrac{3}{\scriptsize\colorTwo{\sin45\degree}}\\[1em] x&=&\dfrac{3}{\sin45\degree}\\[1em] x&\approx&=\dfrac{3}{0.707}\\[1em] x&\approx&4.24 \end{array}$

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

Practice: Solving Equations w/ Primary Trig Ratios

Solve for $x$ in the equation $\cos135\degree=\dfrac{x}{8}$

Round your answer for $x$ to 2 decimal places

I don't know

Practice: Solving Equations w/ Primary Trig Ratios

Solve for $x$ in the equation $\sin150\degree=-\dfrac{5}{x}$

Round your answer for $x$ to 2 decimal places

I don't know

Practice: Primary Trig Ratios

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

Using our calculators, we can see that $\sin30\degree=\dfrac{1}{2}$ .

True or False?

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

I don't know

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

Create a free account to get started

How to Label a Right-Angle Triangle?

Naming a Triangle

The 3 Sides in a Right-Angle Triangle

Practice: Labelling a Right-Angle Triangle

Create a free account to get started

Create a free account to get started

Create a free account to get started

Primary Trigonometric Ratios--Sin, Cos, Tan

How do we remember this?

Calculator Time!

Practice: Primary Trig Ratios

Create a free account to get started

Create a free account to get started

Practice: Primary Trig Ratios

Practice: Primary Trig Ratios

Create a free account to get started

Create a free account to get started

Example: Solving Equations w/ Primary Trig Ratios

Practice: Solving Equations w/ Primary Trig Ratios

Create a free account to get started

Practice: Solving Equations w/ Primary Trig Ratios

Create a free account to get started

Practice: Primary Trig Ratios

Create a free account to get started

Primary Trig Ratios (sin⁡\sinsin, cos⁡\coscos, tan⁡\tantan)

How to Label a Right-Angle Triangle?

Naming a Triangle

The 3 Sides in a Right-Angle Triangle

Practice: Labelling a Right-Angle Triangle

Primary Trigonometric Ratios--Sin, Cos, Tan

How do we remember this?

Calculator Time!

Practice: Primary Trig Ratios

Practice: Primary Trig Ratios

Practice: Primary Trig Ratios

Example: Solving Equations w/ Primary Trig Ratios

Practice: Solving Equations w/ Primary Trig Ratios

Practice: Solving Equations w/ Primary Trig Ratios

Practice: Primary Trig Ratios

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