Simplify \(\dfrac{\sqrt{1+\tan ^2 \theta} \sqrt{1-\sin ^2 \theta}}{\sqrt{\operatorname{cosec}^2 \theta-1}}, \operatorname{cosec}^2 \theta \neq 1\)
- \(\tan \theta\)
- \(\cot \theta\)
- \(\sec \theta\)
- \(1\)
Trigonometry, EXT1 T2 SM-Bank 8
Let `cos (x) = 3/5` and `sin^2(y) = 25/169`, where `x ∈ [{3pi}/{2} , 2 pi]` and `y ∈ [{3pi}/{2} , 2 pi]`.
Find the value of `sin(x) + cos(y)`. (2 marks)
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`text{Both angles are in 4th quadrant (given)}`
`cos(x) = 3/5`
| `sin(x)` | `= – 4/5\ \ text{(4th quadrant)}` |
| `sin^2(y)` | `= 25/169` |
| `sin(y)` | `= – 5/13\ \ text{(4th quadrant)}` |
`cos(y) = 12/13`
| `:. \ sin(x) + cos(y)` | `= – 4/5 + 12/13` |
| `= 8/65` |
Trigonometry, EXT1 T2 EQ-Bank 5
If `sintheta = −4/6` and `−pi/2 < theta < pi/2`,
determine the exact value of `costheta` in its simplest form. (2 marks)
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`text(Consider the angle graphically:)`COMMENT: Pay careful attention to the range of `theta`.
`text(S)text(ince)\ sintheta\ text(is negative) => underbrace(text(4th quadrant))_(−pi/2 <\ theta\ < pi/2)`
`text(Using Pythagoras:)`
`x^2 = 6^2-4^2`
`x = sqrt20 = 2sqrt5`
`:. costheta= (2sqrt5)/6= sqrt5/3`
Trigonometry, EXT1 T2 SM-Bank 4
If `costheta = −3/4` and `0 < theta < pi`,
determine the exact value of `tantheta`. (2 marks)
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`text(Consider the angle graphically:)`
`text(S)text(ince)\ \ costheta\ \ text(is negative ⇒ 2nd quadrant.)`
`text(Using Pythagoras,)`
| `x^2` | `= 4^2 – 3^2` |
| `x` | `= sqrt7` |
`:. tan theta = – sqrt7/3`
Trigonometry, EXT1 T2 2006 HSC 1d
- Simplify `(sin theta + cos theta) (sin^2 theta - sin theta cos theta + cos^2 theta)` (1 mark)
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- Hence, or otherwise, express `(sin^3 theta + cos^3 theta)/(sin theta + cos theta) - 1`, in its simplest form for `0 < theta < pi/2.` (2 marks)
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