Mathematics Module 4: Exponential and Irrational Equations & Trigonometry

Vocabulary flashcards covering exponential equations, irrational equations, unit circle trigonometry, and geometric area calculations based on 1st series Module 4 lecture notes.

Exponential Equations (Equações exponenciais)

Equations where the variable appears as an exponent. They are often solved by making the bases equal, such as converting $128$ to $2^7$ to solve $2^x = 128$.

Auxiliary Variable Method

A technique used to solve complex exponential or higher-order equations by substituting a term with a new variable, such as setting $y = 4^x$ to solve $(4^x)^2 - 5 \times 4^x + 4 = 0$.

Irrational Equations (Equações irracionais)

Equations in which the unknown variable is located under a radical sign. Solving these typically involves raising both sides of the equation to a certain power to eliminate the root.

Even Exponent Rule ($A^n = B^n$)

When the exponent $n$ is even, the equation implies that $A = B$ or $A = -B$, as seen in the resolution of $(2x + 5)^6 = (1 - x)^6$.

Odd Exponent Rule ($A^n = B^n$)

When the exponent $n$ is odd, the equation is equivalent to $A = B$ only, as demonstrated in the case of $(4x + 1)^{13} = (2 - 5x)^{13}$.

Degree to Radian Conversion

The mathematical relationship used to convert angular measures where $180^o$ corresponds to $\pi$ radians.

Smallest Positive Determination (Menor determinação positiva)

The representation of an arc within its first cycle ($0$ to $2\pi$). For a positive arc like $\frac{1435\pi}{3}$, it is found by dividing the numerator by twice the denominator and taking the remainder.

Fundamental Trigonometric Identity

The principle stating that for any angle $\alpha$, the sum of the squares of the sine and cosine is one: $\sin^2(\alpha) + \cos^2(\alpha) = 1$.

Quadrant Signs for Sine (sen x)

Sine is positive ($+$) in the 1st and 2nd quadrants, and negative ($-$) in the 3rd and 4th quadrants.

Quadrant Signs for Cosine (cos x)

Cosine is positive ($+$) in the 1st and 4th quadrants, and negative ($-$) in the 2nd and 3rd quadrants.

Quadrant Signs for Tangent (tg x)

Tangent is positive ($+$) in the 1st and 3rd quadrants, and negative ($-$) in the 2nd and 4th quadrants.

Definition of Tangent (tg x)

The ratio between the sine and the cosine of an angle: $\tan(x) = \frac{\sin(x)}{\cos(x)}$.

Sine of Difference formula

The trigonometric identity used to calculate the sine of the difference between two angles: $\sin(a - b) = \sin(a) \times \cos(b) - \sin(b) \times \cos(a)$.

Standard Triangle Area Formula

Calculated by the product of the base and the corresponding height divided by two: $\text{Area} = \frac{\text{base} \times \text{height}}{2}$.

Heron's Formula

Used to find the area of a triangle given all three sides ($a, b, c$) and the semi-perimeter ($s$): $\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$.

Trigonometric Area Formula

A way to calculate triangle area using two sides and the sine of the included angle: $\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)$, which reaches its maximum value when $\sin(A) = 1$.

Barycenter Property (Baricentro)

The point of intersection of a triangle's medians, which divides each median in a ratio of $2:1$; its distance to a side is one-third of the total height relative to that side.

Area of Similar Triangles

If two triangles are similar with a similarity ratio of $k$, the ratio of their areas is equal to the square of that ratio ($k^2$).