Find the exact values of sin(θ/ 2), cos(θ/ 2), and tan(θ/ 2) for the given conditions. sinθ=(12)

Find the exact values of sin(θ/ 2), cos(θ/ 2), and tan(θ/ 2)...

Key Concepts

Half-Angle Identities

These identities allow you to express trigonometric functions of half angles (such as sin(?/2), cos(?/2), and tan(?/2)) in terms of the trigonometric functions of the full angle ?. They are crucial for transforming and simplifying expressions when the angle is halved, and they typically take the form sin(?/2) = ±?((1 - cos?)/2), cos(?/2) = ±?((1 + cos?)/2), and tan(?/2) = sin?/(1 + cos?) or ±?((1 - cos?)/(1 + cos?)).

Pythagorean Identities

These identities establish the fundamental relationship between sine and cosine, typically expressed as sin²? + cos²? = 1. They are essential for finding an unknown trigonometric function when another is given, by allowing the computation of cosine from a known sine (or vice versa) using the square root of one minus the square of the given value.

Quadrant Sign Considerations

In trigonometry, the sign of a function (positive or negative) depends on the quadrant in which the angle lies. When applying half-angle identities, it is important to choose the correct sign based on the angle’s location in the coordinate system. This concept determines whether the half-angle trigonometric function should take the positive or negative square root, ensuring that the result is consistent with the standard trigonometric signs in their respective quadrants.

Transcript

00:01 In this problem we have given sine theta is equals to 12 upon 13 where theta is greater than 90 degree and smaller than 180 degree and we are asked to find the exact values of sine theta upon 2, cosine theta upon 2 and tangent theta upon 2 so first of all we will find the value of cosine theta as we know that in a right angle triangle the ratio of sine theta is equal to perpendicular upon hypotenuse.

00:38 On comparing to the given values in the equation we get, from here we get perpendicular is equals to 12 and hypotenuse is equal to 13.

00:54 Now we will find the value of base by using the pythagoras theorem, that is, hypotonuse square is equal to perpendicular square plus base square.

01:04 By substituting the value of hypotenuse and perpendicular we get 13 square is equal to 12 square plus base square.

01:17 Now subtracting 12 square on both side we get 13 square minus 12 square is equals to base square.

01:25 From here we get base square is equal to 25.

01:31 Taking the square root we get, base is equal to 5.

01:35 Now we will substitute the value of perpendicular base and base and the square.

01:39 And hypottenues to find the basic trigonometric ratio of cosine theta, that is minus base upon hypotenuse.

01:50 By substituting the value of base and hypotenuse we get cosine theta is equal to minus 5 upon 13.

01:59 Now we will find the value of sine theta upon 2 as we know that sine theta 2 is equal to plus minus under root 1 minus cosine theta upon 2.

02:14 By substituting the value of cosine theta we get sine theta upon 2 is equal to plus minus under root 1 minus minus 5 upon 13 upon 2.

02:31 By simplifying it we get sine theta upon 2 is equal to plus minus under root 13 plus 2 into 13.

02:45 Further we get sine theta upon 2 is equal to plus minus under root 18...

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