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Trigonometry
Trigonometry is a field of mathematics. It is the study of triangles. Trigonometry includes planar trigonometry, spherical trigonometry, finding unknown values in triangles, trigonometric functions, and trigonometric function graphs.
3,810 Questions
What the ecliptic line represent?
The line of the ecliptic is the path that the sun appears to trace through the sky over the period of a year.
How do you solve this logarithmic equation ln x equals 5?
In the equation ln(x) = 5, the solution is x = (about) 148.4. To solve, simply raise e to the power of both sides and reduce...
ln(x) = 5
eln(x) = e5
x = 148.4
Why does sin theta squared plus cos theta squared equal 1?
Sin2(theta) + cos2(theta) = 1 for the same reason that the sides of a right triangle squared equal the hypotenuse squared - The pythagorean theorem.
In the unit circle (origin = (0,0), radius = 1), an angle theta is the angle made by some arbitrary ray drawn from the origin at an angle relative to the x axis. The point of that ray that intersects with the circle is the point (x,y).
Sin(theta) is defined as x, and cos(theta) is defined as y. These are primary trigonometric identities, which link trigonometry with geometry.
Since the points (0,0) (x,0) (x,y) (0,x) describe a right triangle, with (0,x) (0,0) (x,0) being the right angle, then x2 + y2 = 12, or sin2(theta) + cos2(theta) = 1.
If this is not clear, draw a circle around the origin, draw a line from the center to an arbitrary point on the circle, and draw the x and y perpendiculars of that point to each axis. You will see a right triangle. X is sine, Y is cosine, and 1 is hypotenuse. It does not matter if X and/or Y is negative - the squaring will make it positive - and the pythagorean theorem should be visible.
Approx 3.26 km.
How do you solve sin squared theta plus cos theta equals sin theta plus cos squared theta?
For simplicity's sake, X represent theta.
This is the original problem: sin2x+ cosX = cos2X + sinX
This handy-dandy property is key for all you trig fanatics: sin2x+ cos2x = 1
With this basic property, you can figure out that
sin2 x=1-cos2x
and
cos2x= 1-sin2x
So we can change the original problem to:
1-cos2x+cosx = 1-sin2X + sinX
-cos2x + cosx =-sin2x + sinX
Basic logic tells you that one of two things are happening.
sin2x is equal to sinx AND cos2x is equal to cosx. The only two numbers that are the same squared as they are to the first power are 1 and 0. X could equal 0, which has a cosine of 1 and a sine of 0, or it could equal pi/2, which has a cosine of 0 and a sine of 1.
The other possibility whatever x (or theta) is, it's sine is equal to its cosine. This happens twice on the unit circle, once at pi/4 and once at 5pi/4.
If you're solving for all possible values for x and not just a set range on the unit circle, then the final solution is:
x=0+2pin x=pi/2+2pin x= pi/4 +2pin x=5pi/4+2pin (note that n is a variable)
What is the difference between the unit circle and imaginary unit circle?
A unit circle is in the coordinate plane where both axes are measured in real numbers. The imaginary circle is in the complex plane in which one axis (horizontal) measures the real component of a complex number and the other axis measures the imaginary component.
length of the slide= 47.10 feet
This answer uses trigonometry to avoid a lot of work:
tangent = opposite/adjacent and tangent*adjacent (base of ladder from the building) = opposite (height of ladder above ground)
So: tangent 60 degrees*3 = 5.196152423
Therefore: Top of the ladder above ground = 5.2 meters correct to one decimal place.
More laborious methodThe right triangle formed by the wall, ground and ladder has sides in the ratio of 1::2::sq-rt-of-3.The shortest side is the one opposite the 30 degree angle, i.e., the given distance from wall to base of the ladder--3 m.
The length of the ladder represents the hypotenuse of the triangle, and is twice as long, hence 6 m.
And the height of the ladder's top from the ground is proportional to the third side whose length is sq-rt-3 times that of the shortest side. Sq-rt-3 is about 1.732, so height of the ladder's top at the wall is about 5.20 m, or 520 cm.
If you have all sides of a triangle how do you use the law of cosines to solve the angles?
Label the angles of the triangle A, B, and C. Label the side opposite angle A side a, the one opposite angle B side b, and the one opposite angle C side c.
Let's say you want to solve for angle A, you use the law of cosines:
a^2=b^2+c^2-2bcCosA
CosA is the "variable" in this equation, so isolate this. When you have that, you'll have some number (let's call it D) equal to CosA:
D=CosA
Use the inverse Cos function to find the measure of the angle:
Cos^-1(D)=A
And you have the measure of angle A.
From here you can either use the law of cosines again to find a second angle and then the third, though the easier route is usually to just use the law of sines for find the second angle and then the fact that all three angles add to 180 to find the third.
Quadrant II (Quadrant 2) is the region of the coordinate plane (xy-plane, a graph) that is above the x-axis and to the left of the y-axis. In this quadrant, all x values are positive and all y values are negative.
cos (42 degrees) = 0.74314 (rounded)
cos (42 radians) = -0.4 (rounded)
cos (42 grads) = 0.79015 (rounded)
13 feet
i34 is the complex part of the number 0+i34. The real part is 0, so this is a purely imaginary number.
What ratio correctly describes the cosine function?
The cosine of theta is adjacent over hypotenuse, given a right triangle, theta not being the 90 degree angle, adjacent not being the hypotenuse, and theta being the angle between adjacent and hypotenuse.
In a unit triangle, i.e. in a unit circle circumscribed with radius one, and theta and the center of the circle at the origin, cosine of theta is X.
The given angle is 1291 degrees what terminal quadrant does it lie in?
3rd quadrant. The four 90 degree quadrants together formed 360 degrees. When a given angle is greater than 360 degrees, subtract 360 from it till a value smaller than 360 is obtained. In this way, we can determine the quadrant in which the given angle lies. Here the final angle obtained is 211 degrees (1291-3x360=211).
Do I need to learn algebra before trigonometry?
YES! If you can't do algebra, you won't last ten seconds in trigonometry. It basically is algebra, just using equations within equations.
The point (x, y) is moved to (x+pi/4, y).
