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Trigonometry: FAQ

What is trigonometry and why do we need it?

Trigonometry is the study of triangles and the angles and sides that make them. We can use trigonometry to find missing information about triangles, such as how long a side is, or how big an angle is. We can also use trigonometry to model patterns that repeat, such as waves, cycles, and rotations.
Trigonometry is useful in many real-world situations, such as navigation, astronomy, engineering, music, art, and more. For example, we can use trigonometry to find the distance and direction of a ship from a lighthouse, or the height and position of a star in the sky, or the frequency and amplitude of a sound wave, or the shape and symmetry of a design.

What is the unit circle?

The unit circle is a circle with a radius of $1$ centered at the origin. We can use the unit circle to help define the trigonometric functions and visualize their values.

What are inverse trigonometric functions?

Inverse trigonometric functions are the inverse functions of the trigonometric functions. For example, the inverse of the sine function is the arcsine function, written as ${\mathrm{sin}}^{-1}$ or $\mathrm{arcsin}$.

What are the law of sines and law of cosines?

The law of sines and law of cosines are two formulas that can be used to solve any triangles, not just right triangles.
The law of sines states that in any triangle, the ratio of the sine of an angle to the opposite side length is equal for all three angles and sides. It looks like this:
$\frac{\mathrm{sin}\left(A\right)}{a}=\frac{\mathrm{sin}\left(B\right)}{b}=\frac{\mathrm{sin}\left(C\right)}{c}$
Where $A$, $B$, and $C$ are the angle measures of the triangle, and $a$, $b$, and $c$ are the opposite side lengths.
The law of cosines states that in any triangle, the square of a side is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the opposite angle. It looks like this:
$\begin{array}{rl}{a}^{2}& ={b}^{2}+{c}^{2}-2bc\mathrm{cos}\left(A\right)\\ \\ {b}^{2}& ={a}^{2}+{c}^{2}-2ac\mathrm{cos}\left(B\right)\\ \\ {c}^{2}& ={a}^{2}+{b}^{2}-2ab\mathrm{cos}\left(C\right)\end{array}$

What are sinusoidal equations and models?

Sinusoidal equations are equations that contain a sine or cosine function. Sinusoidal models use sinusoidal equations to model periodic phenomena, such as cycles, waves, oscillations, and rotations.

Angle addition identities are formulas that allow us to find the sine or cosine of the sum or difference of two angles. They are useful for simplifying trigonometric expressions, solving trigonometric equations, and proving trigonometric identities.
$\begin{array}{rl}\mathrm{sin}\left(A+B\right)& =\mathrm{sin}\left(A\right)\mathrm{cos}\left(B\right)+\mathrm{cos}\left(A\right)\mathrm{sin}\left(B\right)\\ \\ \mathrm{sin}\left(A-B\right)& =\mathrm{sin}\left(A\right)\mathrm{cos}\left(B\right)-\mathrm{cos}\left(A\right)\mathrm{sin}\left(B\right)\\ \\ \mathrm{cos}\left(A+B\right)& =\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)-\mathrm{sin}\left(A\right)\mathrm{sin}\left(B\right)\\ \\ \mathrm{cos}\left(A-B\right)& =\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)+\mathrm{sin}\left(A\right)\mathrm{sin}\left(B\right)\\ \\ \mathrm{tan}\left(A+B\right)& =\frac{\mathrm{tan}\left(A\right)+\mathrm{tan}\left(B\right)}{1-\mathrm{tan}\left(A\right)\mathrm{tan}\left(B\right)}\\ \\ \mathrm{tan}\left(A-B\right)& =\frac{\mathrm{tan}\left(A\right)-\mathrm{tan}\left(B\right)}{1+\mathrm{tan}\left(A\right)\mathrm{tan}\left(B\right)}\end{array}$

Want to join the conversation?

• wowzers! Great article!
• At what point do we need to learn about secant and cosecant and cotangent identities?
• you need to learn it after completing grade 10 or year 11 or gcse or igcse. It is usually needed for Alevels or college (though you will be taught all of this stuff in your college).
(1 vote)
• For school I had a problem that I got the right answer for this topic, but I can't figure out how it works.
Why does
1 - [(cos²x)/(1 + sinx)]
equal sin(x)?
• cos²(x)=1-sin²(x) by the Pythagorean identity.
1-sin²(x) is a difference of two squares, so you can factor it as (1-sin(x))(1+sin(x)).

Plug this into your expression, and you get
1-[(1-sin(x))(1+sin(x))/(1+sin(x))]
The 1+sin(x) factors cancel, and you're left with

1-[1-sin(x)]

Distribute the negative, and simplify

1-1+sin(x)=sin(x)