For the following exercises, use reference angles to evaluate the expression. If sint=(3)/(4), a

step 1 Hi, so this one is a very challenging problem. We are asked to find a lot of values. We want to

For the following exercises, use reference angles to...

Key Concepts

Reference Angle

A reference angle is the acute angle formed by the terminal side of a given angle and the horizontal axis. It allows you to use known trigonometric values from the first quadrant to determine the values for angles in other quadrants by adjusting for sign differences.

Quadrant and Sign Determination

The quadrant in which an angle terminates determines the sign of its trigonometric functions. For example, in the second quadrant, the sine function is positive while the cosine and tangent functions are negative. This information is essential when transferring reference angle values to the correct context.

Pythagorean Identity

The Pythagorean identity, expressed as sin²? + cos²? = 1, is a fundamental relation in trigonometry. It is used to determine one trigonometric function when another is known, by solving for the unknown value, taking into account the appropriate sign based on the quadrant.

Reciprocal Relationships in Trigonometry

Reciprocal relationships define functions like secant, cosecant, and cotangent in terms of cosine, sine, and tangent respectively. For instance, secant is the reciprocal of cosine, and csc is the reciprocal of sine. These relationships are vital for calculating additional trigonometric values from the primary ones.

Transcript

00:02 Hi, so this one is a very challenging problem.

00:08 We are asked to find a lot of values.

00:11 We want to know cosine t, second t, co -sittantity, and contention t.

00:16 And we are given sine t is equal to 3 over 4.

00:22 And we also know here located in the second quadrant.

00:28 Here, second quadrant.

00:30 Which means sine t will be positive since t located on the positive half of the y -axis.

00:42 And the cosine t will be negative since t located on the negative part of the x -axis.

00:53 And it will be easier for us to solve this problem using the reference angle.

01:00 And we denote the reference angle as t prime.

01:07 And if we draw a line from this point, which is the intersection point between the unircle and the angle, and make the line perpendicular to the x -axis.

01:19 Then we have a triangle with a right angle over here and an angle t prime over here.

01:30 And the hyponius is actually the radius of the unit circle is equal to 1, and the opposite will just be sine t prime, and adjacent will be cosine t prime.

01:50 And since sine t is greater than zero and also located on the positive part of the y -axis, then we know sine t is actually equal to sine t prime.

02:04 And since cosine p is negative, so cosine.

02:11 T is actually equal to negative sine negative cosine prime okay and first we want to find the cosine t and to find a cosine t we can use the reference angle and we can find a cosine prime first and times a negative sign which means we want to find adjacent side, over here is adjacent side, which is also over here, of this triangle.

02:49 And how to find it? we can use the pysogorean theorem.

02:53 We have the hyponius square minus the upper side square, which is sine t prime, which is equal to sine t, which is equal to 3 over 4.

03:04 So 3 over 4 squared equal to adjacent squared.

03:11 And then we have 16 over 16 minus 9 over 16 is equal to 7 over 16 equal to adjacent side square...

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