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3.2 Trigonometry

Tags
Trigonometry
Sine
Cosine
Tangent
Area
Triangle
Sine rule
Cosine rule
True bearings
Angle
Radians
Degrees
Terminology
Definition
Sides of a right triangle
Figure 3.2.1 Right triangle
Sine
sin⁡θ\sin\theta
=OPPHYP=\frac {OPP}{HYP}
sin⁡θ\sin\theta indicates the yy-coordinate of a point on the unit circle, and hence −1≤sin⁡θ≤1-1\leq \sin\theta \leq1.
Cosine
cos⁡θ\cos\theta
=ADJHYP=\frac {ADJ}{HYP}
cos⁡θ\cos\theta indicates the xx-coordinate of a point on the unit circle, and hence −1≤cos⁡θ≤1-1\leq \cos\theta \leq 1.
Tangent
tan⁡θ\tan\theta
=OPPADJ=sin⁡θcos⁡θ=\frac {OPP}{ADJ}=\frac {\sin\theta}{\cos\theta}
tan⁡θ\tan \theta refers to the gradient of a line from the origin to a point on the unit circle.
Inverse trigonometric ratios
Inverse trigonometric functions find the angle that corresponds to a given trigonometric ratio. They are denoted as: sin⁡−1x\sin^{-1}x or arcsin⁡x\arcsin x cos⁡−1x\cos^{-1}x or arccos⁡x\arccos x tan⁡−1x\tan^{-1}x or arctan⁡x\arctan x
Applications
Angle of elevation
The angle formed by the horizontal line and the line of sight when an observer looks at an object above the horizontal level.
Angle of depression
The angle formed by the horizontal line and the line of sight when an observer looks at an object below the horizontal level.
True bearings
A way of describing direction, measured clockwise from the north direction.
It is expressed in degrees, ranging from 0° to 360°.
Figure 3.2.2 True bearings
Trigonometric geometry:
1.
Area A=12absin⁡CA = \frac12ab\sin C
2.
Sine rule: asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} =\frac {b}{\sin B}=\frac {c}{\sin C}
3.
Cosine rule: c2=a2+b2−2absin⁡Cc^2 = a^2+b^2-2ab\sin C
Special Angles: π2\fracπ2, π4\frac π4, π6\fracπ6, 00, ππ (needs to be memorized)
3.2 Trigonometry
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3.2 Trigonometry