Find exact values of the six trigonometric functions for each angle. Rationalize denomi. nators

Find exact values of the six trigonometric functions for...

Key Concepts

Coterminal Angles

Coterminal angles are angles that share the same terminal side after rotation. By adding or subtracting full rotations (360° or 2? radians), any given angle can be adjusted to an equivalent angle within a standard interval, which simplifies the process of finding trigonometric function values.

Rationalizing Denominators

Rationalizing denominators involves the process of rewriting expressions so that no radicals appear in the denominator. This common algebraic technique is often applied when expressing exact values of trigonometric functions to ensure that the results are in their simplest and most accepted form.

Reference Angles

A reference angle is the acute angle formed by the terminal side of an angle and the horizontal axis. Using a reference angle allows for the determination of trigonometric function values in different quadrants by relating them back to the known acute angle values, with adjustments made for sign based on the quadrant.

Trigonometric Functions

Trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant define relationships between the angles and sides of a right triangle or correspond to the coordinates of a point on the unit circle. Their exact values for special angles are often expressed in radical form and are essential tools in solving geometric and trigonometric problems.

Unit Circle

The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate system, where the coordinates of any point on the circle correspond to the cosine and sine values of the angle formed. This concept is crucial for understanding and determining the values of trigonometric functions at various angles.

Transcript

0:00 Hello everyone.

00:02 Here we have to find the trigonometric ratios for negative 855 degree.

00:07 That is theta is negative 855 degree.

00:14 Since this is a negative one, we will first convert it into a positive value of theta, which would be equivalent to this one.

00:22 And such type of pairs or of angles are called co -terminal angles.

00:29 Okay.

00:30 So we have to find the co -terminal angles.

00:32 Terminal angle of negative 855 degree and this co -terminal angle should be the lowest positive value one okay so for that we add multiples of 360 degree now which multiple of 360 degree should be added to negative 855 degree such that the value becomes the lowest possible positive value you can see the second multiple of 360 degrees 720 degree which is less than 855 so that would not be a positive value then we can add the third multiple of 360 degree which is equal to negative 855 degree plus 1 080 degree okay so if you subtract 855 from 1080 you will get 225 degree so that is positive 225 degree so this is the lowest positive co -terminal value of negative 855 degree now we are at negative 225 degree which is not the angle we want to find the thermometric ratios we need the smallest angles which are in the first quadrant that means the value which are known to us either 30 degree 45 degree or 60 degree.

02:30 So these angles are known to us, the value of these trigromatic ratios for these angles.

02:36 So we have to find an angle which is called a reference angle.

02:42 And that reference angle would be used to find the trigromatic ratios.

02:51 And we have to also remember that the reference angle of negative 85 degree would be equal to the reference angle of 225 degrees.

03:00 Because they're coterminal angles that means they're equivalent so to find the reference of 225 degree we will first check in which quadrant it actually is so the first quadrant ends at 90 degree the second one at 180 degree and the third one at 270 degree so that means 225 degree would be on the third quadrant so here we write three so that we remember this is in the third quadrant...

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