Essential Mathematics: Trigonometry
29 flashcards · Mathematics · by lakesare
What's 1 rad approximately in degrees?
360°2π\frac{360°}{2π}2π360° = 57.295°
What's the length of the arc?
arc length = rθ.
What's the area of sector?
area of sector = 1/2 r2θ.
[Area of sector, arc length] (a) The shape is itself a sector – of what angle? (b) What is the total perimeter of the shape? Give your answer in cm, to one decimal place. (c) What is the area of the shape? Give your answer in cm2, to one decimal place. Classic computer game character :-)
How to find, via geometry:
Using the equilateral triangle with sides of length 2 & pythagoras theorem. Let's begin with the angle π/3 (60◦ ). This angle is one of the angles of an equilateral triangle, that is, a triangle in which all angles are equal. This is because π/3 (60◦ ) is a third of the total angle sum π (180◦ ) of a triangle.Figure shows an equilateral triangle that is split into two right-angled triangles by a vertical line. The equilateral triangle has sides of length 2, and the vertical line divides the bas…
How to find, via geometry:
the square & pythagoras theorem. It is created by splitting a square down one of its diagonals. Because each angle of a square is a right angle, π/2, each of the acute angles of the triangle is half of a right angle, π/4.
Show why this is true:
π/2 - θ is the opposite acute angle, which turns around cosine and sine :-o
Imagine graphs of sin(θ) and cos(θ).
What should we do to the graph of sin(θ) to get the graph of cos(θ)?
revert across y-axis add pi/2 to x - that is, translate x by -pi/2
Sin, cos, and tan are not one-to-one, - which means their inverse functions should be restricted.What are the accepted restricted ranges of sin cos tan?
Here is an already-restricted graph of sin(x). How does the graph of arcsin(x) look?
Arcsin is an inverse of sin(x) - which means it's reflected across the x=y line.
Prove the sine rule.
Find without using the calculator.
I like to use the unit circle in combination with recalling ranges of sin-1, however here is what OU offers to use:All the values of sine, cosine and tangent in the special angles table are positive, so you can’t use this table directly to find the exact values of the inverse sine, inverse cosine or inverse tangent of any negative numbers. However, you can find these values by using symmetry properties of the inverse trigonometric functions that follow from the symmetry properties of the trigon…
If we apply sin−1 to solve this equation: sin θ = 12\frac{1}{2}21 We get a single value, - which's fair, arcsin being a function θ = sin−1( 12\frac{1}{2}21 ) = π/6. However, as indicated in the graph, there are other solutions. How to find other solutions?
Find 2 values in 2π, & then add 2π to each of them. // todo - what about tan-s?
Is there some difference between -π/6 & 2π - π/6?
There is! E.g. if x∈[-π/2, π/2].
Find all solutions between π and 3π of the equation sin θ = −12-\frac{1}{2}−21 .
Find all solutions in the interval [−π, π] of the equation sin θ = 0.
−π, 0 and π. The equation sin θ = 0 tells you that the associated point P on the unit circle with angle θ has y-coordinate 0. That is, P lies on the x-axis, so the solutions in [−π, π] are −π, 0 and π.
Find all solutions between −180° and 180° of the equation tan θ = 10. Give your answers to the nearest degree.
Find the angle C, in degrees to one decimal place.
There are two possible solutions: C = 164.053°, and C = 15.946°. Former is impossible, so the answer is: C = 15.9°, to one decimal place. So if you use the sine rule to find an unknown angle, then you may only be able to narrow down the answer to two possibilities: an acute angle, and an obtuse angle that is equal to 180◦ minus the acute angle. You can find the correct answer only if you have other information that tells you whether the unknown angle is acute or obtuse.
Give area of triangle in terms of θ, a and b.
area = 12\frac{1}{2}21 ab sin(θ).
Explain these in terms of a graph.
The next set of identities comes from other symmetry properties of the graphs of the trigonometric functions. The vertical axis is a line of mirror symmetry for the graph of cosine, which means that cos θ and cos(−θ) are equal, for every angle θ. This gives the identity cos(−θ) = cos θ. However, the sine of −θ is equal to the negative of the sine of θ, as you can see from Figure 60(a). That is, sin(−θ) = − sin θ, for every angle θ. There is a similar identity for tangent.
What's cosec(θ)?
cosec θ = 1sin(θ)\frac{1}{\sin(θ)}sin(θ)1
What's sec(θ)?
sec θ = 1cos(θ)\frac{1}{\cos(θ)}cos(θ)1
What's cot(θ)?
cot θ = 1tan(θ)\frac{1}{\tan(θ)}tan(θ)1
What's sec(θ), cosec(θ), and cot(θ)?
Prove that sin(A + B) = sin(A) cos(B) + sin(B) cos(A)
Area of the large triangle = 1st area + 2nd area. Use the A = sin(θ)pq/2 formula to find all areas.
What's a simple proof for sin(2θ) = 2 sin θ cos θ?
Using sin(A + B) = sin A cos B + cos A sin B with the same angle.
What's a simple proof for cos(2θ) = cos2θ − sin2θ?
Using cos(A + B) = cos A cos B - sin A sin B with the same angle.
See the graphs of the functions f(t) = sin(20t) and g(t)=sin(22t). They represent waves, such as sound waves, that are almost the same but have slightly different periods. Now consider what happens when you add these two functions. You obtain the function h(t) = sin(20t) + sin(22t), Why does 'adding' these two similar waves produces a wave with an interesting shape, - consisting of fast oscillations with a slow variation in the magnitude of these oscillations.
So the graph of the function h can be thought of as the rapidly oscillating graph of sin(21t), scaled by the factor 2 cos(t), which produces a slow variation in magnitude, known as beating. h(t) = sin(20t) + sin(22t) = 2 sin(21t) cos(t). You can observe this phenomenon when two musical notes with similar frequencies are played simultaneously, and the resulting slow variation in magnitude is heard as 'beats'.