Recap: Trigonometry And Triangles
When we talked about opens in a new windowthe world of trigonometry, we learned that the part of math called trigonometry deals with triangles. And, in particular, its the part of math that deals with figuring out the relationship between the three sides and the three angles that make up every triangle.
Of particular interest to us is the special type of triangles known as right triangles. Every right triangle has one 90-degree angle , and two angles that each range between anything larger than 0 degrees and smaller than 90 degrees .
For our discussion of sine, cosine, and tangent , its important that we have a way of labeling the sides of right triangles.
As we learned opens in a new windowlast time, the longest side of a triangle is known as its hypotenuse. The side opposite the angle were looking at is known as the opposite side . And the side adjacent to the angle were looking at is known as the adjacent side.
Graph Of The Sine Function
The graph of sine is periodic, meaning that it repeats itself indefinitely and has a domain of -< x< . The sine graph has an amplitude of 1 its range is -1y1. Below is a graph of y=sin in the interval , showing just one period of the sine function.
Repeating this portion of y=sin indefinitely to the left and right side would result in the full graph of sine. Below is a graph showing four periods of the sine function in the interval .
In this graph, we can see that y=sin exhibits symmetry about the origin if the graph is reflected about the origin, it creates a mirror image. This confirms that sine is an odd function, since -sin=sin.
Table Showing Different Values Of Sin Cos And Tan With Respect To Radians
1. Evaluate the value of Sin 90 + Cos 90.
Sol. As we know that the value of Sin 90 = 1
And the value of Cos 90 = 0
Substituting the values of Sin 90 and Cos 90 ,
Therefore, Sin 90 + Cos 90 = 1 + 0
2. What is the value of Sin 270 + 2 Tan 45.
Sol. As we know that the value of Sin 270 = -1
And the value of Tan 45 = 1
Substituting the values of Sin 270 and Tan 45 ,
Therefore, Sin 270 + 2Tan 45 = -1+2×1 = -1+2 = 1
Read Also: What Is Red Shift In Physics
The Basic Two: Sine And Cosine
A picture is worth a thousand words . The trig functions are nothing more thanlengths of various sides of a right triangle in various ratios.Since there are three sides, there are 3 × 2 = 6different ways to make a ratio of sides. Thatswhy there are six trig functions, no more and no less.
Of those six functions, threesine, cosine, andtangentget the lions share of the work.Well begin with sine and cosine, because they are really basicand the others depend on them.
Here is one of the conventional ways of showinga right triangle. A key point is that the lower-case letters a, b, care the sides opposite to the angles marked with the correspondingcapital letters A, B, C. Most books use this convention:lower-case letter for side opposite upper-case angle.
The two fundamental definitions are marked in the diagram.You must commit them to memory.In fact, they should become second nature toyou, so that you recognize them no matter how the triangle is turnedaround. Always, always, the sine of an angle is equal to the oppositeside divided by the hypotenuse . The cosine isequal to the adjacent side divided by the hypotenuse .
sine = / hypotenuse
cosine = / hypotenuse
What is the sine of B in the diagram? Remember opp/hyp: the oppositeside is b and the hypotenuse is c, sosin B = b/c.What about the cosine of B? Remember adj/hyp: the adjacent side isa, so cos B = a/c.
sin A = cos or cos
cos A = sin or sin
What Are Sine Cosine And Tangent
What do sines, cosines, and tangents have to do with right triangles? And what are the “sin,””cos,” and “tan” buttons on your calculator for? Keep reading to find out!
- The ratios of the sides of a right triangle are completely determined by its angles.
- Because there are three sides of a triangle means that there are also three possible ratios of the lengths of a triangle’s sides.
- These three ratios are the sine, cosine, and tangent trigonometric functions.
Do you know what two angles living inside the same right triangle said to each other? The first angle goes, Hey Thelma , I dont mean to go off on a tangent here, but whats your sine? To which the second angle replies, Phil , I dont know why you even bother to ask, my sine is obviously the same as your cosine!
Okay, so maybe thats not the best joke in the world, but once you understand sines and cosines, it is kind of funny. Of course, that means that if you dont know the difference between a sine and a cosine, youre currently left out in the metaphorical cold.
Clearly we cant let that happenand we wont! Because today were going to learn all about sines, cosines, and tangents.
Also Check: What Are The Branches Of Chemistry
Sine Function Domain And Range
As we know, the sine function is defined for all real numbers, so the domain of y = sin x is the set of all real numbers, i.e. R. The range of sine function is the closed interval . That means, -1 y 1 or -1 sin x 1. However, the range of this function can be given as per the quadrants. Observe the below table to get the range of sine function in different quadrants.
|-1 < sin < 0|
A Little Information About What A Right
Let us consider a right-angle triangle named ABC, with its three sides namely the opposite, adjacent, and the hypotenuse. In a right-angled triangle, we generally refer to the three sides according to their relation with the angle. The little box in the right corner of the triangle given below denotes the right angle which is equal to 90°.
The 3 sides of a right-angled triangle are as follows-
The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse .
The side that is opposite to the angle is known as the opposite .
The side which lies next to the angle is known as the Adjacent
Pythagoras theorem states that,
In a right-angled triangle,
As our angle of interest is Sin 0. So accordingly, the Sin function of an angle or Sin 0 Degrees is equal to the ratio of the length of the opposite side to the length of the hypotenuse .
Recommended Reading: Do Video Game Designers Use Math
Ratios For Sum Angles
As the examples showed, sometimes we need angles other than 0, 30, 45, 60, and 90 degrees. In this chapter you need to learn two things:1. Sin is not equal to sin A + sin B. It doesn’t work like removing the parentheses in algebra.2. The formula for what sin does equal.
First to show that removing parentheses doesn’t “work.” Here: make A 30 degrees and B 45 degrees.
Sin 30 is 0.5. Sin 45 is 0.7071. Adding the two is 1.2071.
You know that no sine can be more than 1. Why? the ratio has the hypotenuse as its denominator. The most that the numerator can be is equal to the denominator. A sine or cosine can never be greater than 1, so a value of 1.2071 must be wrong.
What Is The Value Of Sin 0
A little about Trigonometry and Trigonometric Ratios-
Trigonometric ratios in Trigonometry are derived from the three sides of a right-angled triangle, basically the hypotenuse, the base , and the perpendicular .
According to the trigonometric ratio in maths, there are three basic or primary trigonometric ratios also known as trigonometric identities.
To be more specific, they are used in right-angled triangles, the triangles with one angle equal to 90 Degrees
Read Also: What Is Undefined In Math
Sine Cosine And Tangent
With all of these preliminaries now happily splashing around inside our growing pool of mathematical knowledge, were finally ready to tackle the meaning of sine, cosine, and tangent. Heres the key idea:
The ratios of the sides of a right triangle are completely determined by its angles.
The ratios of the sides of a right triangle are completely determined by its angles.
In other words, the value you get when you divide the lengths of any two sides of a right trianglelets say the length of the side opposite one of its angles divided by its hypotenuseis entirely set in stone as soon as the angles are set in stone.
Why? Well, if the angles are fixed, making the triangle bigger or smaller has no impact on the relative lengths of its sides. But changing the triangles angles, even a tiny bit, does! If you need some convincing, try drawing a few triangles of your own and youll see that it is indeed true.
Now, the fact that there are three sides of a triangle means that there are also three possible ratios of the lengths of a triangles sides. And, as you might have already guessed, these three ratios are none other than the famous sine, cosine, and tangent trigonometric functions.
The Other Four: Tangent Cotangent Secant Cosecant
The other four functions have no real independent life of their own theyre just combinations of the first two. You could do all oftrigonometry without ever knowing more than sines and cosines. But knowingsomething about the other four, especially the tangent, can often save you somesteps in a calculationand your teacher will expect you to knowabout them for exams.
I find it easiest to memorize the definition of the tangentin terms of the sine and cosine:
tan A = /
Youll use the tangent function a lot morethan the last three functions.
Theres an alternative way to remember the meaning of the tangent.Remember from the diagramthat sin A = opposite/hypotenuse and cos A =adjacent/hypotenuse. Plug those into equation 3, the definition ofthe tan function, and you have tan A = / or
tangent = /
Notice this is not marked memorize: you dont have tomemorize it because it flows directly from the definitionequation 3, and in fact the two statements areequivalent. Ive chosen to present them in this order to minimize thejumble of opp, adj, and hyp among sin, cos, andtan. However, if you prefer you can memorizeequation 4 and then derive the equivalent identityequation 3 whenever you need it.
Example: A guy wire is anchored in the ground and attachedto the top of a 45-foot flagpole. How far is the anchor from the baseof the flagpole, if the wire meets the ground at anangle of 63°?
tan A = a/b
b = a/tan A = 45/tan 63° =about 22.9 ft.
sin A = a/cc = a/sin A and
Recommended Reading: What Is Fk In Physics
Why Do Cosecant And Secant Have The Same Period
Cosecant and se- cant have the same period as sine and cosine do, namely 2. Cotangent has period , just as tangent does. In terms of formulas, the previous two sentences mean that csc = csc sec = sec cot = cot Its easy to check why these functions have the periods that they do.
How to check the identity of cosecant and cotangent?
Cosecant and cotangent are odd functions, meaning that csc = csc and cot = cot. We can check that these identities are true by using that sine is an odd function and that cosine is even: csc = 1 sin = 1 sin = csc cot = cos sin = cos sin = cot 275. Cofunction identities.
Properties Of Sine Function As Per Quadrants
The sine function has values positive or negative depending upon the quadrants. In the above table, it is seen that sine 270 is negative while sine 90 is positive. For the sine function, the value depends upon the quadrants and is positive for the first and second quadrants while it is negative for values in the third and fourth quadrants.
Four quadrants in Trigonometry diagram:
Don’t Miss: Discovering Geometry Chapter 3 Answers
What Is Cos And Sin Actually Doing
I am having the hardest time figuring out what sin and cos are doing when you enter in calculator.
What I do understand about them
1) They are both essentially finding the max and min values for their respective axis. cos being x and sine being y.
-this makes PERFECT sense. Yay!
2) You can find what other lengths of a triangle are. ) ) **most cases theta needs to be in radians i understand that.
-this makes sense. Yay!
Ok, so what the heck is going on when you do sin. How does that equal 1/2? I understand that basically the sine of $\pi$/6 would be 1/2, ok but why? when you do cos, x = 1 because r = 1 . This is where I am confused. Again, I can do this stuff on calculator but I am a programmer and if things don’t make sense my head spins and I need to understand what is going on.
Expressions For Lengths Of Sides
The definitions of sine and cosine can be rearranged a little bit tolet you write down the lengths of the sides in terms of thehypotenuse and the angles. For example, when you knowthat b/c = cos A, youcan multiply through by cand get b = c × cos A. Can you write anotherexpression for length b, one that uses a sine instead of a cosine?Remember that opposite over hypotenuse equals the sine, sob/c = sin B. Multiplythrough by c and you haveb = c × sin B.
Can you see how to write down two expressions for the length of sidea? Please work from the definitions and verify thata = c × sin A =c × cos B.
Example: Given, a right triangle with angle A =52°and hypotenuse c = 150 m. What is the length of sideb? Hint: draw a picture, and label A, c,and b.
Solution: Pictures are always good. You donthave to obsess over getting the picture exactly right, but at leastmake it close. That will help you see when your answer is impossible,so you know youve made a mistake. In my little sketch, I setout to make angle A a bit more than 45°, but to my eyesit looks like a bit less. Thats okay.
You may notice that I marked side a, even though wedont need it for the problem. I did that so I didnt haveto think about which side was b. Always remember the rulethat the side with a given letter is opposite the angle with thatletter.
Once you have the picture, solving the problem is prettystraightforward. You want something involving A, itsadjacent side, and the hypotenuse that has to be the cosine.
Don’t Miss: What Does Vertices Mean In Math
Six Functions In One Picture
You saw earlier how the sine and cosineof an angle are the sides of a triangle in a unit circle. It turns outthat all six functions can be shown geometrically in this way.
unit circle sin = BC cos = AC tan = EDcsc = AG sec = AE cot = FG
Graphic courtesy ofTheMathPage
In the illustration at right, triangle ABC has angle at thecenter of a unit circle . You alreadyknow that BC = sin and AC = cos .
What about tan ? Well, since DE is tangent to the unitcircle, you might guess that its length is tan , and youd beright. Triangles ABC and AED are similar, and therefore
ED / AD = BC / AC
ED / 1 = sin / cos
ED = tan
More information comes from the same pair of similartriangles:
AE / AB = AD / AC
AE / 1 = 1 / cos
AE = sec
The lengths that represent cot and csc will comefrom the other triangle, GAF. That triangle is also similar totriangle AED.
Using similar triangles GAF and AED,
FG / FA = AD / ED
FG / 1 = 1 / tan
FG = cot
That makes sense: FG is tangent to the unit circle, and is thetangent of the complement of angle , namely angle GAF.Therefore, FGis the cotangent of the original angle .
Finally, using the same pair of similar triangles again, youcan also say that
AG / FA = AE / ED
AG / 1 = sec / tan
AG = /
AG = 1 / sin
AG = csc
This one diagram beautifully depicts the geometrical meaningof all six trig functions when the angle is drawn at the center of aunit circle:
sin = BC cos = AC tan = ED
Ratios For Difference Angles
Now, you have two ways to obtain formulas for difference angles. First, use a geometric construction, such as the one that was used for sum angles, reversing it so that is the angle B subtracted from the angle A.
In reasoning similar to that which was used for the sum angles, presented here somewhat abbreviated, are the sine and cosine formulas:
Also Check: What Is Reproduction In Biology
Java Math Sin Method With Examples
The java.lang.Math.sin returns the trigonometry sine of an angle in between 0.0 and pi. If the argument is NaN or infinity, then the result is NaN. If the argument is zero, then the result is a zero with the same sign as the argument. The value returned will be between -1 and 1.
public static double sin
Parameter: The value whose sine is to be returned.
Return type: This method returns the sine value of the argument.
Here we will be proposing 2 examples one in which we will simply be showcasing the working of Math.sin method of java.lang packagemethod and secondary be edge case of the first example specific taken where argument is NaN or infinity.
Trigonometry Sine Cosine And Tangent
ABC is a right angled triangle
The angle A is 30 degrees. We write this as:
a is the symbol for the side opposite angle A
b is the symbol for the side opposite angle B
c is the symbol for the side opposite angle C
Similar triangles are triangles in which all the angles in one triangle are equal to the angles in the other triangle
These two triangles are similar. The ratio between two sides in one triangle is equal to the ratio between the corresponding sides in the other triangle.
Using the notation in the above triangles we get the following:
The ratio depends on the size of the angle.
The ratio called tangent of an acute angle in a right angled triangle is defined as the ration between the side opposite the angle and the side adjacent to the angle .
Don’t Miss: Chapter 9 Test Form 2a Answers Geometry