Geometry: Inverse Sine and Cosine
This is a free lesson from our course in Geometry
 
   
In this lesson you’ll learn the basics and concepts of Inverse Cosine and Sine, First the basics will be reviewed, which will follow to discuss the important relationship, and reference trigonometric ratios that help to apply them to solve real world application problems. The presentation covering such content will be done by the instructor in own handwriting, using video and with the help of several examples and solution. This will help you understand important concepts and relationships to solve home work problems and also use them for real life applications. (More text below video...)
<h2> Geometry - Inverse Sine and Cosine - Watch video</h2> <p> angle, sine, tangent, cosine, sine ratio, cosine ratio, opposite, adjacent, hypotenuse, tangent ratio, geometry, sine inverse, example, cosine inverse, SOHCAHTOA, geometry help, solution, practice questions, quizzes </p> <p> The inverse trigonometric functions allow you to find the measure of an angle in a right triangle. All that you need to know are any two sides as well as how to use SOHCAHTOA.</p>
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(Continued from above) It explains now the general difference between  sine and inverse or sin-1: sine is the ratio of two actual sides of a right triangle (the opposite and hypotenuse), say
  sin () = AC/AB
Inverse or sin,-1 is an operation that uses these two sides of a right triangle like sine does (opposite over hypotenuse), but for finding the measure of the angle, say
  sin-1 (AC/AB) = measure of angle ()
 
Notice that both sine and inverse sine in a right triangle , involve the opposite side and hypotenuse of a right triangle, the result of these two operations is different i.e. where as (sine) finds the ratio of these two sides; sine inverse actually calculates the measure of the angle.
A review from earlier learning indicates that standard trig functions are periodic, i.e. they repeat themselves. Therefore, you may find the same output value for multiple input values of the function. Notice that to define an inverse function, the original function must be one-to-one i.e. each value in the domain must correspond to exactly one value in the range, and vice-versa.
Further on to define now the inverse functions for sine and cosine, the domains of these functions are restricted (Fig: below). These restricted function(s) are called Sine and Cosine . Note the capital ‘S’ and ‘C’ in Cosine.
The inverse sine function is defined as the inverse of the restricted Sine function y = Sin x i.e. Sin-1 (sin x) = x – /2 x   /2. Similarly the inverse cosine function is defined as the inverse of the restricted Cosine function Cos-1(cos x) = x x . Therefore,
Identities for the inverse sine and cosine are respectively:
Sin-1 (sin x)  = x – /2 x /2
Cos-1(cos x)  = x x

As stated above, measure of angles will be found out from the known information about the values of the related trigonometric functions. All you may be needed to do is to use calculator or refer to the trig tables including values of angles. To get more clear understanding on the relationship of angle and their references, see the diagram below:
Say, a circle having radius of 1 unit and an angle in standard position. A is the point where the angle intersects the circle at [(3/2), 1/2]and B is the point on x-axis represented by (1, 0). With this information mA is to be found out?
You know that opp = (3/2) and hyp = 1(radius of the circle). There fore sin(A) = opp/hyp= (3/2). Angle with this sin is 60 (Refer trig table). This can also be done using calculator like:
  
In other notation, it can be expressed as: sin-1(3/2) = mA = 60. It is common to say ‘the inverse sine of (3/2) is 60. Alternatively you may express ‘the angle with a sine of (3/2) is 60.

The variation of the values of sine and Cosine
• cos decreases from 1 to 0, as increases from 0 to /2
• sin increases from 0 to 1, as increases from 0 to /2
• the values of cos decrease from 0 to -1, as increases from /2 to
• sin increases from 1 to 0, as increases from /2 to
• the values of cos increase from -1 to 0, as increases from to 3/2 
• sin decreases from 0 to -1, as increases from to 3/2
• the values of cos increase from 0 to 1, as increases from 3/2 to 2
• sin increases from -1 to 0, as increases from 3/2 to 2

Look into some of the illustrative examples given below:

Example 1: Given- an angle has a sine of – 1/2. Find out the measure of the angle?
Step 1: sine of the angle is – 1 /2 (given)
Step 2: You know the measure of angle is sin-1 (1/2) = 30.
Notice that it is with negative sign, this angle could be either in quadrant three or four.
Step 3: If in quadrant three, it could be either 210 or (210 - 360) = - 150.
Step 4: If in quadrant four, it could be -30 or (360 - 30) = 330
The measure of angle could be: 210, -150, -30, 330, as the final answer.

Example 2: Given- an angle has cos() = 0.5. What is the measure of this reference angle ().
Step 1: cos () = 0.5. (Given)
Step 2: You may recall from earlier learning on the relationship of sine and cosine for special triangle 30- 60- 90. In such a case, adj of the angle is 1, and hyp is 2.
Step 3: Refer to the trig table or use calculator.
It gives 60.
The measure of angle in the reference is 60, as the final answer.
The video above will explain more in detail about Inverse Cosine and Sine, and how to apply the concepts in solving real-world problems. This is explained with the help of several examples and done watching video. This helps you to deal with solving problems and help doing the Geometry home work.
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