Question

3) State the signs of cos 4 and cos4º. Which of
these two functions is greater ?​

Answer 1

Answer:

We define the trigonometric functions for angles greater than 90° in the following way:

initial side

terminal side

θ

r

x

y

(x, y)

x-axisy-axis

An obtuse angle in standard position.

By Pythagoras, \displaystyle{r}=\sqrt{{{x}^{2}+{y}^{2}}}r=

x

2

+y

2

. Then the ratios are:

\displaystyle \sin{\theta}=\frac{y}{{r}}sinθ=

r

y

\displaystyle \cos{\theta}=\frac{x}{{r}}cosθ=

r

x

\displaystyle \tan{\theta}=\frac{y}{{x}}tanθ=

x

y

\displaystyle \csc{\theta}=\frac{r}{{y}}cscθ=

y

r

\displaystyle \sec{\theta}=\frac{r}{{x}}secθ=

x

r

\displaystyle \cot{\theta}=\frac{x}{{y}}cotθ=

y

x

How is this different to the definitions we already met in section 2, Sine, Cosine, Tangent and the Reciprocal Ratios? The only difference is that now x or y (or both) can be negative because our angle can now be in any quadrant. It follows that the trigonometric ratios can turn out to be negative or positive. In the earlier section, the angles involved were always less than 90° so all 6 ratios were positive.

Notice that r is always positive.

Example 1

Let's see how the trigonometric ratios are defined using a particular example. Let our angle θ be defined by the point \displaystyle{\left(-{2},{3}\right)}(−2,3) in the following way: