04Dfindingsides

toc = Trigonometry =

Finding Side Lengths
Recall that the basic rules for trigonometry are:

To find the length of side:
 * identify which two sides are involved in the problem
 * identify which of the 3 trig rules includes those two sides
 * write down the relevant trig rule
 * substitute the information that is known
 * rearrange to get the pronumeral on its own
 * {this is called making the pronumeral the subject}
 * use your calculator to find the value
 * check your calculator is in DEGREES mode
 * round to a sensible number of decimal places
 * write the answer and include units if available

** Example 1 **
__**Solution:**__
 * (a) ** Find the value of x in the triangle shown:

{The two sides involved are OPP and HYP so use __**sin**__}

math \\ . \qquad \sin \big( \theta \big) = \dfrac{\text{OPP}}{\text{HYP}} \\ \\ \\ . \qquad \sin \big( 22^\circ \big) = \dfrac{x}{7} \qquad \{ \times 7 \} \\ \\ . \qquad x = 7 \times \sin \big( 22^\circ \big) \\ \\ . \qquad x = 2.62 \text{ cm} math

__**Solution:**__
 * (b) ** Find the value of x in the triangle shown:

{The two sides involved are ADJ and HYP so use __**cos**__}

math \\ . \qquad \cos \big( \theta \big) = \dfrac{\text{ADJ}}{\text{HYP}} \\ \\ \\ . \qquad \cos \big( 57^\circ \big) = \dfrac{x}{9.3} \qquad \{ \times 9.3 \} \\ \\ . \qquad x = 9.3 \times \cos \big( 57^\circ \big) \\ \\ . \qquad x = 5.07 \text{ m} math

** Example 2 **

 * (a) ** Find the value of x in the triangle shown:

__**Solution:**__ {The two sides involved are OPP and ADJ so use __**tan**__}

math \\ . \qquad \tan \big( \theta \big) = \dfrac{\text{OPP}}{\text{ADJ}} \\ \\ \\ . \qquad \tan \big( 35^\circ \big) = \dfrac{15.2}{x} \qquad \qquad \{ \times \; x \} \\ \\ . \qquad x \times \tan \big( 35^\circ \big) = 15.2 \qquad \{ \div \tan \big( 35^\circ \big) \} \\ \\ . \qquad x = \dfrac{15.2}{\tan \big( 35^\circ \big) } \\ \\ \\ . \qquad x = 21.71 \text{ cm} math

__**Solution:**__
 * (b) ** Find the value of x in the triangle shown:

{The two sides involved are OPP and HYP so use __**sin**__}

math \\ . \qquad \sin \big( \theta \big) = \dfrac{\text{OPP}}{\text{HYP}} \\ \\ \\ . \qquad \sin \big( 73^\circ \big) = \dfrac{9}{x} \qquad \qquad \{ \times \; x \} \; and \; \{ \div \sin \big( 73^\circ \big) \} \\ \\ . \qquad x = \dfrac{9}{\sin \big( 73^\circ \big) } \\ \\ \\ . \qquad x = 9.41 \text{ mm} math


 * NOTE: **
 * In Eg 1, the pronumeral was the __numerator__ (top of the fraction) and we ended up __multiplying__
 * In Eg 2, the pronumeral was the __denominator__ (bottom of the fraction) and we ended up __dividing__

Checking Answers
You can check your answer for how sensible it is:
 * the __hypotenuse__ should be the __longest__ side
 * the size of the angle should guide you about the expected side length

If your answer doesn't seem sensible, you may have made a mistake.

in Eg 1A (above)
 * For example: **
 * The hypotenuse is 7cm, so x should be shorter than that.
 * The angle is 22º, which is fairly small (smaller than what is shown in the diagram)
 * so we expect the answer to be much smaller than 7cm
 * Actual answer was 2.62cm so that seems sensible.

For another site that explains this idea, go here: MathIsFun

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