/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Evaluate (if possible) the six t... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 25

Evaluate (if possible) the six trigonometric functions at the real number. $$t=4 \pi / 3$$

Short Answer

Expert verified
\[ \sin \frac{4 \pi}{3} = - \frac{\sqrt{3}}{2}, \cos \frac{4 \pi}{3} = -\frac{1}{2}, \tan \frac{4 \pi}{3} = \sqrt{3}, csc \frac{4 \pi}{3} = - \frac{2}{\sqrt{3}}, sec \frac{4 \pi}{3} = -2, cot \frac{4 \pi}{3} = \frac{1}{\sqrt{3}} \]

Step by step solution

01

Determine the quadrant

The given input \( t = \frac{4 \pi}{3} \) is more than \( \pi \) (180掳) and less than \( 2\pi \) (360掳) which means it lies in the third quadrant. In this quadrant, sine and cosine are negative, while tangent is positive.
02

Find the reference angle

The reference angle in the third quadrant is found by subtracting \( \pi \) from the given angle. So, the reference angle is \( \frac{4 \pi}{3} - \pi = \frac{ \pi}{3} \) which corresponds to 60掳 in degrees. The values of sin, cos and tan for 60掳 are \( \frac{\sqrt{3}}{2} \), \( \frac{1}{2} \) and \( \sqrt{3} \) respectively.
03

Calculate the trigonometric functions

Now we'll calculate the values based on Steps 1 and 2. \n For sine: \( \sin{\frac{4\pi}{3}} = - \sin{\frac{\pi}{3}} = -\frac{\sqrt{3}}{2} \).\n For cosine: \( \cos{\frac{4\pi}{3}} = -\cos{\frac{\pi}{3}} = -\frac{1}{2} \).\n For tangent: \( \tan{\frac{4\pi}{3}} = \tan{\frac{\pi}{3}} = \sqrt{3} \).\n The values of cosec, sec and cot are reciprocal of sin, cos and tan respectively. Hence, the values are \(-\frac{2}{\sqrt{3}}, -2, \) and \( \frac{1}{\sqrt{3}} \) respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometry in Radians
When it comes to evaluating trigonometric functions, understanding the radian measure is crucial.
Unlike degrees, which divide a circle into 360 equal parts, radians provide a direct connection between the angle size and the arc length it subtends. One complete revolution of a circle is equal to approximately 6.28 radians, which is equivalent to 360 degrees, and this relation comes from the circumference formula of a circle, where the circumference is given by 2蟺谤 (with r being the radius).

A radian measure can be more handy, especially when dealing with mathematics at higher levels, like calculus, because it simplifies many formulas. For example, when you are given an angle in radians, like 迟=4蟺/3, you鈥檙e being told directly that the arc length for a unit circle (radius r=1) is 4蟺/3 units long. This is why in advanced mathematics and physics, radians are often the preferred unit of measurement for angles.
Reference Angle
A reference angle is perhaps one of the handiest tools in trigonometry when it comes to finding function values.
The reference angle is the smallest angle that the terminal side of the given angle makes with the x-axis. What makes the reference angle so useful is that it always falls between 0 and 蟺/2 radians (or 0 and 90 degrees), and thus corresponds to an angle in the first quadrant where all trigonometric functions are positive.

To find it, you subtract multiples of 蟺/2 if your angle is in radians, or 90 degrees if it's in degrees, depending on the quadrant the angle lies in. For example, if you have an angle like 迟=4蟺/3 (which lies in the third quadrant), you subtract 蟺 from it, resulting in the reference angle 蟺/3. The value of sine, cosine, and tangent for the reference angle can be directly used to find the function values for the original angle with proper sign adjustment according to the quadrant.
Unit Circle Quadrant Analysis
The unit circle is a powerful tool that offers insight into how angles relate to trigonometric functions. This circle has a radius of one unit and is centered at the origin of a coordinate plane. Each point on the circle corresponds to an angle, measured from the positive x-axis.
Quadrant analysis involves determining which of the four quadrants an angle's terminal side falls in when in its standard position. Here鈥檚 a quick guide:
  • First Quadrant: Angles between 0 and 蟺/2 radians 鈥� all trigonometric functions are positive.
  • Second Quadrant: Angles between 蟺/2 and 蟺 radians 鈥� sine is positive, cosine and tangent are negative.
  • Third Quadrant: Angles between 蟺 and 3蟺/2 radians 鈥� tangent is positive, sine and cosine are negative.
  • Fourth Quadrant: Angles between 3蟺/2 and 2蟺 radians 鈥� cosine is positive, sine and tangent are negative.

Once you know in which quadrant your angle resides, you can easily deduce the signs of your trigonometric function values. Using the angle 迟=4蟺/3 as an example, we determined from quadrant analysis that sine and cosine must be negative and tangent must be positive, which allows us to assign correct signs to the values we obtained from the reference angle 蟺/3.

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