From SOH CAH TOA

`sin = frac(text(opposite))(text(hypotenuse))`

`cos = frac(text(adjacent))(text(hypotenuse))`

Divide Sin by Cos

`= frac(text(opposite))(text(hypotenuse)) ÷ frac(text(adjacent))(text(hypotenuse))`

`= frac(text(opposite))(text(hypotenuse)) xx frac(text(hypotenuse))(text(adjacent))`

`= frac(text(opposite))(text(adjacent))`

which is Tan. Therefore the Tan of an angle `= frac(text(sin of that angle))(text(cos of that angle))`.

Pythagoras` Theorem is given by `a^2 + b^2 = c^2`

Divide throughout by `c^2` and simplify

`(frac(a)(c))^2 + (frac(b)(c))^2 = 1`

And replacing *adjacent* ÷ *hypotenuse* with sin and *opposite* ÷ *hypotenuse* with cos:

`(sin x)^2 + (cos x)^2 = 1`

Normally this is written without the brackets, but to ensure that you obtain the sine of the angle before squaring, it is written as

`sin^2 x + cos^2 x = 1`

(`sin x^2` would mean square the angle then take the sine).

If `tan theta = frac(sin^2 theta)(cos theta)`, then what is the value of `theta` given 0 < `theta` < 360?

Given | `tan theta` | `= frac(sin^2 theta)(cos theta)` |

Using | `frac(sin theta)(cos theta)` | `= tan theta` |

substituting | `frac(sin theta)(cos theta)` | `= frac(sin^2 theta)(cos theta)` |

multiply by `cos theta` | `sin theta` | `= sin^2 theta` |

divide by `sin theta` | `1` | `= sin theta` |

`sin-1` both sides | `90` | `= theta` |

Answer: 90º

For 0 < `theta` < 90, what is the value of `theta` when `sin^2 theta = cos theta`? Give your answer to the nearest whole degree.

Given | `sin^2 theta` | `= cos theta` |

Using | `sin^2 theta + cos^2 theta` | `= 1` |

substitute for `sin^2theta` | `cos theta + cos^2 theta` | `= 1` |

rearrange | `cos^2 theta + cos theta - 1` | `= 0` |

let `cos theta = x` | `x^2 + x - 1` | `= 0` |

solve the quadratic | `x` | `= 1.618 or 0.618` |

substitute back | `cos theta` | `= 1.618 or 0.618` |

not possible: | `cos theta` | `= 1.618` |

valid answer: | `cos theta` | `= 0.618` |

cos^{-1} both sides |
`theta` | `= 51.8` |

nearest degree | `theta` | `= 52` |

Answer: 52º

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