Find the exact values of the trigonometric functions for the acute angle θ. sec θ=(6)/(5)

Name: Problem 21, Chapter 5: Precalculus: Functions and Graphs
Uploaded: 2021-05-02T16:51:09-08:00
Duration: 4 min 11 s
Channel: Raushan Kumar
Description: Find the exact values of the trigonometric functions for the acute angle θ. sec θ=(6)/(5)

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

Key Concepts

Pythagorean Identity

The Pythagorean identity, sin²? + cos²? = 1, is a fundamental relationship in trigonometry that binds the sine and cosine functions. It is used to calculate one trigonometric function when the other is known, ensuring consistency in the behavior of trigonometric ratios for all angles.

Right Triangle Trigonometry

Right triangle trigonometry involves defining trigonometric functions as ratios of the sides of a right triangle. This concept is used to derive exact trigonometric values and to solve problems involving acute angles by constructing a reference triangle that embodies the given ratios.

Reciprocal Relationships

In trigonometry, reciprocal identities relate functions such as cosine and secant, where sec ? is the reciprocal of cos ?. This concept allows one to find one function from the other, and it is essential for converting between different trigonometric expressions.

Exact Trigonometric Values

Finding exact values in trigonometry means expressing the trigonometric ratios in their simplest, non-decimal form, often involving radicals. This approach is crucial for achieving precise answers in problems requiring analytical solutions.

Transcript

00:01 We are going to do problem number 21 and this question the the expression for sac theta is given sec theta is given that is six by five so we need to find the expression for another technometric functions all technometric functions okay so we will understand that what we can conclude when this information is given so let's say this is the triangle and this is height this is base and this is hypotenous so if sate theta is given this simply means that hypotenius by base is given okay so we have the value of hypotenius we have the value of base so we can easily find the value of height okay by pythagoras theorem pythagoras theorem is hypotenous square equals to base square plus height square so here height is square will be a hypotenous square minus base square so height square will be hypotenous here is six so six is square that is 36 and basis square that is 25 five is 25 now here is square will not be there because we just take it to the right side and put it this under root under root sign so square sign will not be there so from here height is coming out to be 11 under root of 11 so we got the height here now we can easily write we got height also so let me write here we got height also that is under root 11 now you can easily write the value of alternative function so let us start by writing sign theta so this is equals to height by hypotenous so height we got is under root of 11 by hypotaneous we have is 6.

02:09 So let us just find the value of under root 11 and this is divided by this is divided by 6.

02:20 So this is equals to this value is equal to 0 .553.

02:29 Now cossack theta.

02:33 So cossack theta is simply the inverse of sine so this is 1 by sine theta.

02:38 So this will be equals to 1 by 0 .553 so let us just find the value that 1 divide by 0 .553 this is equals to 1 .808 now cost theta cost theta this will be just the opposite of this sec theta okay opposite of sec theta so this will be 5 by 6 so let us just find the value of 5 by 6 so this is 0 .833 now we have 10 theta and cot theta to find so 10 theta so 10 theta is height by base so height here is under root 11 and base here is 5 so let us just find the value under root 11 divide by 5 so this is giving us 0 .6 now cot theta, cot theta is the inverse of 10 theta, that is 1 by 10 theta...

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