### 数学代写|微积分代写Calculus代写|The values of the six trig functions

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## 数学代写|微积分代写Calculus代写|The values of the six trig functions

The values of the six trig functions are
$$\begin{array}{ll} \sin \theta=\frac{-12}{13} & \csc \theta=\frac{13}{-12} \ \cos \theta=\frac{5}{13} & \sec \theta=\frac{13}{5} \ \tan \theta=\frac{-12}{5} & \cot \theta=\frac{5}{-12} . \end{array}$$
Developing the habit of writing the trig function values in three rows and two columns as presented in example 3 is helpful. When the ratios in the first column are applied, the values in the second column are written quickly as the reciprocals of those in the first column.
No matter which point on the terminal side of the angle is used (figure 14), the values of the trig functions are the same. This is important, for we would not want two different values for $\sin \theta$, as that would make it fail to be a function. Because the results are the same no matter which point is chosen, we say that the trig functions are well-defined.

Two “special triangles” from geometry are helpful for quickly determining the values of trig functions: the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle and the $30^{\circ}-60^{\circ}-90^{\circ}$ triangle. With angles labeled in radians, these triangles are pictured in figure 15 . To use these triangles effectively, they must be on instant recall.

## 数学代写|微积分代写Calculus代写|Finding additional trig values

Given the value of one trig function of an angle and the quadrant in which the angle lies, the values of the other trig functions can be determined as well.

Example 9 Given $\sin \theta=\frac{4}{7}$ and $\frac{\pi}{2}<\theta<\pi$, find the other five trig functions of $\theta$.

Solution There are always two quadrants in which a given trig function has positive values. Knowing that $\sin \theta$ is a positive number is not enough to know in which quadrant the terminal side of $\theta$ lies.

We are told that the angle $\theta$ is a second-quadrant angle $(\theta$ is between $\frac{\pi}{2}$ and $\pi$. We begin by drawing an angle with the terminal side in the second quadrant, then we choose a point on the terminal side and drop a perpendicular to form a right triangle (figure 21 , left).
Figure 21 Steps in drawing the diagram for example 9
Since $\sin \theta=\frac{y}{r}=\frac{4}{7}$, we label the vertical leg of the triangle 4 and the hypotenuse 7 (figure 21 , right). The label for the horizontal leg, $x$, should be negative because we are in the second quadrant.

The value of $x$ may be found using the Pythagorean theorem:
\begin{aligned} x^{2}+4^{2} &=7^{2} \ x^{2}+16 &=49 \ x^{2} &=33 \ x &=-\sqrt{33} . \end{aligned}
We finish the diagram by labeling the horizontal leg (figure 22 ).
Figure 22 The completed figure for example 9
We are now ready to determine the values of the other five trig functions using the definition:
$$\begin{array}{ll} \cos \theta=\frac{-\sqrt{33}}{7} & \csc \theta=\frac{7}{4} \ \tan \theta=\frac{4}{-\sqrt{33}} & \sec \theta=\frac{7}{-\sqrt{33}} \ \cot \theta=\frac{-\sqrt{33}}{4} . \end{array}$$

## 数学代写|微积分代写Calculus代写|Useful trig identities

There are a very large number of trigonometric identities, equations that relate trig functions to one another and are true for all values for which the functions are defined. Some identities are much more commonly used than others, and the ability to recognize the applicability of an identity readily can be crucial to completing some calculus exercises. The identities described next are the ones that all calculus students should know.

The first set of identities includes the reciprocal identities. Since $\sin \theta=\frac{y}{r}$ and $\csc \theta=\frac{r}{y}$, the two trig functions are reciprocals of one another.

Notice that
$$\frac{\sin \theta}{\cos \theta}=\frac{\frac{y}{x}}{\frac{x}{r}}=\frac{y}{x}=\tan \theta .$$
This is called a ratio identity.
RATIO IDENTITIES
For any angle $\theta$ for which the functions are defined,
$$\tan \theta=\frac{\sin \theta}{\cos \theta} \quad \cot \theta=\frac{\cos \theta}{\sin \theta}$$
The reciprocal identities for secant and cosecant along with the ratio identities for tangent and cotangent give us the ability to write every trigonometric function in terms of sines and cosines. This ability can be very helpful in certain calculus exercises.

Example 10 Rewrite the expression $\tan \theta \csc \theta+2 \sec \theta$ in terms of sines and cosines.
Solution Using the reciprocal and ratio identities,
\begin{aligned} \tan \theta \csc \theta+2 \sec \theta &=\frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\sin \theta}+2 \cdot \frac{1}{\cos \theta} \ &=\frac{1}{\cos \theta}+\frac{2}{\cos \theta} \ &=\frac{3}{\cos \theta} . \end{aligned}

## 数学代写|微积分代写Calculus代写|Finding additional trig values

X2+42=72 X2+16=49 X2=33 X=−33.

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