Trigonometry

Lesson

As with every topic in mathematics, there is a conceptual side (what you need to know and understand) and a practical side (what you need to do and answer). To calculate values involving trigonometric expressions, it will often be easiest to use a scientific calculator.

Don't forget the three trigonometric ratios!

Trigonometric ratios

$\sin\theta=\frac{\text{Opposite }}{\text{Hypotenuse }}$`s``i``n``θ`=Opposite Hypotenuse

$\cos\theta=\frac{\text{Adjacent }}{\text{Hypotenuse }}$`c``o``s``θ`=Adjacent Hypotenuse

$\tan\theta=\frac{\text{Opposite }}{\text{Adjacent }}$`t``a``n``θ`=Opposite Adjacent

If $\sin\theta=0.65$`s``i``n``θ`=0.65, find $\theta$`θ` to the nearest degree.

**Think:** This question is asking us what the angle ($\theta$`θ`) is, if the ratio of the opposite side and hypotenuse is $0.65$0.65. To answer this question, we can use the inverse sine button on a calculator. It will probably look like $\sin^{-1}$`s``i``n`−1, and may involving pressing 'shift' or '2nd F'.

**Do:**

$\sin\theta$sinθ |
$=$= | $0.65$0.65 | |

$\theta$θ |
$=$= | $\sin^{-1}\left(0.65\right)$sin−1(0.65) |
(Take the inverse sine) |

$\theta$θ |
$=$= | $40.54160187$40.54160187$\ldots$… | (Evaluate with a calculator) |

$\theta$θ |
$=$= | $41$41 | (Round to the nearest degree) |

If $\cos\theta=0.146$`c``o``s``θ`=0.146, find $\theta$`θ`, writing your answer to the nearest degree.

If $\sin\theta=1$`s``i``n``θ`=1, find $\theta$`θ`.

If $\tan\theta=1.711$`t``a``n``θ`=1.711, find $\theta$`θ`, writing your answer to the nearest degree.

Determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem