Here are examples of reciprocal trig function transformations: $$\displaystyle y=-{{\sec }^{{-1}}}\left( {\frac{x}{3}} \right)-\frac{\pi }{2}$$. Since we want sec of this angle, we have $$\displaystyle \sec \left( \theta \right)=\frac{r}{x}=\frac{{17}}{8}$$. If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software. This function has a period of 2π because the sine wave repeats every 2π units. Worked Example. Graphing trig functions can be tricky, but this post will talk you through some of the tips and tricks you can use to be accurate every single time! Tangent is not defined at these two points, so we can’t plug them into the inverse tangent function. So, be careful with the notation for inverse trig functions! As shown below, we will restrict the domains to certain quadrants so the original function passes the horizontal lin… We can also write trig functions with “arcsin” instead of $${{\sin }^{-1}}$$: if  $$\arcsin \left( x \right)=y$$, then $$\sin \left( y \right)=x$$. Next we limit the domain to [-90°, 90°]. Then use Pythagorean Theorem $$\left( {{{{\left( {-12} \right)}}^{2}}+{{y}^{2}}={{{13}}^{2}}} \right)$$ to see that $$y=5$$. The main differences between these two graphs is that the inverse tangent curve rises as you go from left to right, and the inverse cotangent falls as you go from left to right. Remember that when functions are transformed on the outside of the function, or parentheses, you move the function up and down and do the “regular” math, and when transformations are made on the inside of the function, or parentheses,  you move the function back and forth, but do the “opposite math”: $$\displaystyle y={{\sin }^{{-1}}}\left( {2x} \right)-\frac{\pi }{2}$$. eval(ez_write_tag([[300,250],'shelovesmath_com-large-mobile-banner-1','ezslot_9',127,'0','0']));eval(ez_write_tag([[300,250],'shelovesmath_com-large-mobile-banner-1','ezslot_10',127,'0','1']));eval(ez_write_tag([[300,250],'shelovesmath_com-large-mobile-banner-1','ezslot_11',127,'0','2']));IMPORTANT NOTE: When getting trig inverses in the calculator, we only get one value back (which we should, because of the domain restrictions, and thus quadrant restrictions). We can transform and translate trig functions, just like you transformed and translated other functions in algebra. In inverse trig functions the “-1” looks like an exponent but it isn’t, it is simply a notation that we use to denote the fact that we’re dealing with an inverse trig function. Graph is moved up $$\displaystyle \frac{\pi }{4}$$ units. In other words, the inverse cosine is denoted as $${\cos ^{ - 1}}\left( x \right)$$. Example Questions. We still have to remember which quadrants the inverse (inside) trig functions come from: Note:  If the angle we’re dealing with is on one of the axes, such as with the arctan(0°), we don’t have to draw a triangle, but just draw a line on the $$x$$ or $$y$$-axis. This trigonometry video tutorial explains how to graph secant and cosecant functions with transformations. By Sharon K. O’Kelley . You can also put trig composites in the graphing calculator (and they don’t have to be special angles), but remember to add $$\pi$$ to the answer that you get (or 180° if in degrees) when you are getting the arccot or $${{\cot }^{{-1}}}$$ of a negative number (see last example). Purplemath. They can be used to find missing sides or angles in a triangle, but they can also be used to find the length of support beams for a bridge or the height of a tall object based on a shadow. Examples of special angles are 0°, 45°, 60°, 270°, and their radian equivalents. Especially in the world of trigonometry functions, remembering the general shape of a function’s graph goes a long way toward helping you remember more […] When solving trig equations, however, we typically get many solutions, for example, if we want values in the interval $$\left[ {0,2\pi } \right)$$, or over the reals. ), $$\displaystyle -\frac{\pi }{4}$$ or  –45°, $$\displaystyle \frac{{5\pi }}{6}$$ or  150°. 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