5.1 Algebra with Trigonometric Ratios

In this chapter we apply some techniques from algebra to analyze more complicated trigonometric expressions. Before we begin, let's review some algebraic terminology.

Note

Algebra Refresher

An algebraic expression is any meaningful collection of numbers, variables, and operation symbols. For example, the height of a golf ball is given in feet by the expression $- 16 t^{2} + 64 t$ , where $t$ is the number of seconds after the ball is hit.
We evaluate an expression by substituting a specific value for the variable or variables involved. Thus, after 1 second, the height of the golf ball is
$- 16 (1)^{2} + 64 (11) = - 16 + 64 = 48 feet$
and after 2 seconds, the height is
$- 16 (2)^{2} + 64 (2) = - 64 + 128 = 64 feet$
and so on.

Evaluating Trigonometric Expressions

Trigonometric ratios represent numbers, and they may appear as part of an algebraic expression. Expressions containing trig ratios can be simplified or evaluated like other algebraic expressions.

Example 1

Evaluate each expression for $X equals 30 degrees$ and $Y equals 135 degrees$ .

$2 tan ⁡ open parenthesis Y close parenthesis plus 3 sin ⁡ open parenthesis X close parenthesis$
$6 tan ⁡ open parenthesis X close parenthesis cos ⁡ open parenthesis Y close parenthesis$

Substituting the values for $X$ and $Y$ , we get
$2 tan ⁡ open parenthesis 135 degrees close parenthesis plus 3 sin ⁡ open parenthesis 30 degrees close parenthesis$
Next, we evaluate each trig ratio and follow the order of operations.
$\begin{aligned} 2 \tan (135^{\circ}) + 3 \sin (30^{\circ}) & = 2 (- 1) + 3 (\frac{1}{2}) \\ = - 2 + \frac{3}{2} = \frac{- 1}{2} \end{aligned}$
This expression includes the product of two trig ratios, $\tan (X) \cos (Y)$ .
$\begin{aligned} 6 \tan (X) \cos (Y) & = 6 \tan (30^{\circ}) \cos (135^{\circ}) \\ = 6 (\frac{1}{\sqrt{3}}) (\frac{- 1}{\sqrt{2}}) \\ = \frac{- 6}{\sqrt{6}} = - \sqrt{6} \end{aligned}$

Evaluate each expression for $X equals 30 degrees$ , $Y equals 60 degrees$ .

$4 sin ⁡ open parenthesis 3 X plus 45 degrees close parenthesis$
$1 minus cos ⁡ open parenthesis 4 Y close parenthesis$

$2 the square root of 2$
$\frac{3}{2}$

Simplifying Trigonometric Expressions

Because expressions such as $\sin (x)$ and $\tan (θ)$ are just variables, we can use algebra skills to simplify expressions involving the trig functions. For the rest of this section, we'll try to illustrate all the skills you will need going forward with your study of trigonometry.

Note

More Algebra

When we simplify an algebraic expression, we obtain a new expression that has the same values as the old one, but is easier to work with. For example, we can apply the distributive law and combine like terms to simplify

$\begin{aligned} 2 x (x - 6) + 3 (x + 2) & = 2 x^{2} - 12 x + 3 x + 6 \\ = 2 x^{2} - 9 x + 6 \end{aligned}$

The new expression is equivalent to the old one, that is, the expressions have the same value when we evaluate them at any value of $x$ . For instance, you can check that, at $x equals 3$ ,

$\begin{aligned} 2 (3) (3 - 6) + 3 (3 + 2) & = 6 (- 3) + 3 (5) = - 3 \\ and 2 (3)^{2} - 9 (3) + 6 & = 18 - 27 + 6 = - 3 \end{aligned}$

To simplify an expression containing trig ratios, we treat each ratio as a single variable. Compare the two calculations below:

$\begin{aligned} 8 x y & - & 6 x y & = 2 x y \\ 8 \cos (θ) \sin (θ) & - & 6 \cos (θ) \sin (θ) & = 2 \cos (θ) \sin (θ) \end{aligned}$

Both calculations are examples of combining like terms. In the second calculation, we treat $\cos (θ)$ and $\sin (θ)$ as variables, just as we treat $x$ and $y$ as variables in the first calculation.

Simplify

$2 \cos (t) - 4 \cos (w) \sin (w) + 3 \cos (t) - 2 \cos (w)$

$5 \cos (t) - 4 \cos (w) \sin (w) - 2 \cos (w)$

Example 3

Simplify, and evaluate for $z equals 40 degrees$ .

$3 \sin (z) - \sin (z) \tan (z) + 3 \sin (z)$

We can combine like terms to get

$6 sin ⁡ open parenthesis z close parenthesis minus sin ⁡ open parenthesis z close parenthesis tan ⁡ open parenthesis z close parenthesis$

Because $40 degrees$ is not one of the angles for which we know exact trig values, we use a calculator to evaluate the expression.

$\begin{aligned} 6 \sin (40^{\circ}) - \sin (40^{\circ}) \tan (40^{\circ}) & = 6 (0.6428) - (0.6428) (0.8391) \\ = 3.3174 \end{aligned}$

Simplify, and evaluate for $x = 25^{\circ}, y = 70^{\circ}$

$3 \cos (x) + \cos (y) - 2 \cos (y) + \cos (x)$

$3.2832$

Powers of Trigonometric Ratios

Compare the two expressions

$(\cos (θ))^{2} and \cos (θ^{2})$

They are not the same.

The first expression, $(\cos (θ))^{2}$ , says to compute $\cos (θ)$ and then square the result.
$\cos (θ^{2})$ says to square the angle first, and then compute the cosine.

For example, if $θ equals 30 degrees$ , then

$\begin{aligned} (\cos (30^{\circ}))^{2} & = {(\frac{\sqrt{3}}{2})}^{2} = \frac{3}{4} \\ but \cos ((30^{2})^{\circ}) & = \cos (900^{\circ}) = \cos (180^{\circ}) = - 1 \end{aligned}$

We usually write $\cos^{2} (θ)$ instead of ${(\cos (θ))}^{2}$ , to distinguish it from $\cos (θ^{2})$ , and to reduce the number of parentheses. Thus, $\cos^{2} (θ)$ means the square of $\cos (θ)$ .

The same notation applies to the other trig ratios, so that

$\sin^{2} (θ) = (\sin (θ))^{2} and \tan^{2} (θ) = (\tan (θ))^{2}$

Other powers are written in the same fashion. Thus, for example, $\sin^{3} (θ) = (\sin (θ))^{3}$ .

Evaluate $tan to the power 4 ⁡ open parenthesis 60 degrees close parenthesis$

$9$

Products

We can multiply together trigonometric expressions, just as we multiply algebraic expressions. Recall that we use the distributive law in computing products such as

$x (3 x - 2) = 3 x^{2} - 2 x$

and

$(x - 3) (x + 5) = x^{2} + 2 x - 15$

Example 5

Using the distributive law, multiply $cos ⁡ open parenthesis t close parenthesis open parenthesis 3 cos ⁡ open parenthesis t close parenthesis minus 2 close parenthesis$ .

Think of $\cos (t)$ as a single variable, and multiply by each term inside parentheses. (The algebraic form of the calculation is shown below in blue).

$\begin{aligned} \cos (t) (3 \cos (t) - 2) & = (\cos (t)) (3 \cos (t)) - (\cos (t)) \cdot 2 \\ = 3 \cos^{2} (t) - 2 \cos (t) \\ x (3 x - 2) & = x \cdot 3 x - x \cdot 2 \\ = 3 x^{2} - 2 x \end{aligned}$

Notice that we write $(\cos (t))^{2}$ as $\cos^{2} (t)$ .

Multiply $2 tan ⁡ open parenthesis β close parenthesis open parenthesis 4 tan squared ⁡ open parenthesis β close parenthesis plus tan ⁡ open parenthesis α close parenthesis close parenthesis$

$8 tan cubed ⁡ open parenthesis β close parenthesis plus 2 tan ⁡ open parenthesis β close parenthesis tan ⁡ open parenthesis α close parenthesis$

We can also use the distributive law to multiply binomials that include trig ratios. You may have used the acronym $F O I L$ to remember the four multiplications in a product of binomials:

$F$ irst terms, $O$ utside terms, $I$ nside terms, and $L$ ast terms.

Example 6

Multiply $(4 \sin (C) - 1) (3 \sin (C) + 2)$ .

This calculation is similar to the product $(4 x - 1) (3 x + 2)$ , except that the variable $x$ has been replaced by $\sin (C)$ . Compare the calculations for the two products; first the familiar algebraic product:

$\begin{aligned} (4 x - 1) (3 x + 2) & = 4 x \cdot 3 x + 4 x \cdot 2 - 1 \cdot 3 x - 1 \cdot 2 \\ = 12 x^{2} + 8 x - 3 x - 2 \\ = 12 x^{2} + 5 x - 2 \end{aligned}$

We compute the product in this example in the same way, but replacing $x$ by $\sin (C)$ .

$\begin{aligned} (4 & \sin (C) - 1) (\sin (C) + 2) \\ = (4 \sin (C)) (3 \sin (C)) + (4 \sin (C)) \cdot 2 - 1 (3 \sin (C)) - 1 \cdot 2 \\ = 12 \sin^{2} (C) + 5 \sin (C) - 2 \end{aligned}$

Expand $open parenthesis 4 cos ⁡ open parenthesis α close parenthesis plus 3 close parenthesis squared$

$16 \cos^{2} (α) + 24 \cos (α) + 9$

Factoring

We can factor trigonometric expressions with the same techniques we use for algebraic expressions. In the next two Examples, compare the familiar algebraic factoring with a similar trigonometric expression.

Factor.

$2 a squared minus a b$
$2 cos squared ⁡ open parenthesis ϕ close parenthesis minus cos ⁡ open parenthesis ϕ close parenthesis sin ⁡ open parenthesis ϕ close parenthesis$

$a open parenthesis 2 a minus b close parenthesis$
$cos ⁡ open parenthesis ϕ close parenthesis open parenthesis 2 cos ⁡ open parenthesis ϕ close parenthesis minus sin ⁡ open parenthesis ϕ close parenthesis close parenthesis$

We can also factor quadratic trinomials.

Example 8

Factor.

$t^{2} - 3 t - 10$
$\tan^{2} (α) - 3 \tan (α) - 10$

We look for numbers $p$ and $q$ so that $(t + p) (t + q) = t^{2} - 3 t - 10$ . We see that their product must be $p q = - 10$ , and their sum must be $p + q = - 3$ . By checking the factors of $minus 10$ for the correct sum, we find $p = - 5$ and $q equals 2$ . Thus,
$t^{2} - 3 t - 10 = (t - 5) (t + 2)$
Now replace $t$ by $\tan (α)$ to find
$\tan^{2} (α) - 3 \tan (α) - 10 = (\tan (α) - 5) (\tan (α) + 2)$

Factor.

$3 z^{2} - 2 z - 1$
$3 \sin^{2} (β) - 2 \sin (β) - 1$

$(3 z + 1) (z - 1)$
$(3 \sin (β) + 1) (\sin (β) - 1)$

Review the following skills you will need for this section.

Section 5.1 Summary

Vocabulary

Expression
Evaluate
Binomial
Trinomial
Simplify
Equivalent expression
Like terms
Distributive law
Factor

Concepts

Expressions containing trig ratios can be simplified or evaluated like other algebraic expressions. To simplify an expression containing trig ratios, we treat each ratio as a single variable.
$\sin (X + Y)$ is not equal to $\sin (X) + \sin (Y)$ (and the same holds for the other trig ratios). Remember that the parentheses indicate function notation, not multiplication.
We write $\cos^{2} (θ)$ to denote $open parenthesis cos ⁡ 9 close parenthesis squared$ , and $\cos^{n} (θ)$ to denote $(\cos (θ))^{n}$ . (Similarly for the other trig ratios.)
We can factor trigonometric expressions with the same techniques we use for algebraic expressions.

Study Questions

To evaluate $cos squared ⁡ open parenthesis 30 degrees close parenthesis$ , Delbert used the keystrokes
$COS 30 x squared ENTER$
and got the answer 1. Were his keystrokes correct? Why or why not?
Make up an example to show that $\tan (θ + ϕ) \neq \tan (θ) + \tan (ϕ)$ .
Factor each expression, if possible.
1. $x squared minus 4 x$
2. $x squared minus 4$
3. $x squared plus 4$
4. $x^{2} + 4 x + 4$
5. $x^{2} - 4 x + 4$
6. $x squared plus 4 x$
7. $x^{2} - 4 x - 4$
8. $- x^{2} + 4$

Skills

Evaluate trigonometric expressions #1–22
Simplify trigonometric expressions #23–34
Recognize equivalent expressions #35–44
Multiply or expand trigonometric expressions #45–56
Factor trigonometric expressions #57–70

Homework 5.1

For Problems 1–8, evaluate the expressions, using exact values for the trigonometric ratios.

$5 tan ⁡ open parenthesis 135 degrees close parenthesis plus 6 cos ⁡ open parenthesis 60 degrees close parenthesis$

$minus 2$

$3 tan ⁡ open parenthesis 240 degrees close parenthesis plus 8 sin ⁡ open parenthesis 300 degrees close parenthesis$

$sin ⁡ open parenthesis 15 degrees plus 30 degrees close parenthesis$

$\frac{1}{\sqrt{2}}$

$cos ⁡ open parenthesis 2 times 75 degrees close parenthesis$

$8 cos squared ⁡ open parenthesis 30 degrees close parenthesis$

$6$

$12 sin squared ⁡ open parenthesis 315 degrees close parenthesis$

$3 tan squared ⁡ open parenthesis 150 degrees close parenthesis minus sin squared ⁡ open parenthesis 45 degrees close parenthesis$

$\frac{1}{2}$

$1 plus tan squared ⁡ open parenthesis 120 degrees close parenthesis$

For Problems 9–16, evaluate the expressions for $x = 30^{\circ}, y = 45^{\circ}$ , and $z equals 60 degrees$ . Give exact values for your answers.

$3 sin ⁡ open parenthesis x close parenthesis plus 5 cos ⁡ open parenthesis y close parenthesis$

$4$

$4 tan ⁡ open parenthesis y close parenthesis plus 6 cos ⁡ open parenthesis y close parenthesis$

$minus 2 tan ⁡ open parenthesis 3 y close parenthesis$

$2$

$sin ⁡ open parenthesis 3 z minus 2 x close parenthesis$

$\cos^{2} (x) + \sin^{2} (x)$

$1$

$7 sin squared ⁡ open parenthesis y close parenthesis plus 7 cos squared ⁡ open parenthesis y close parenthesis$

$\cos (x) \cos (z) - \sin (x) \sin (z)$

$0$

$tan ⁡ open parenthesis 180 degrees minus x close parenthesis tan ⁡ open parenthesis x close parenthesis$

For Problems 17–22,evaluate the expressions using a calculator.

$sin ⁡ open parenthesis 10 degrees plus 40 degrees close parenthesis$
$sin ⁡ open parenthesis 10 degrees close parenthesis plus sin ⁡ open parenthesis 40 degrees close parenthesis$
$sin ⁡ open parenthesis 10 degrees close parenthesis cos ⁡ open parenthesis 40 degrees close parenthesis plus cos ⁡ open parenthesis 10 degrees close parenthesis sin ⁡ open parenthesis 40 degrees close parenthesis$

$0.7660$
$0.8164$
$0.7660$

$cos ⁡ open parenthesis 20 degrees plus 50 degrees close parenthesis$
$cos ⁡ open parenthesis 20 degrees close parenthesis plus cos ⁡ open parenthesis 50 degrees close parenthesis$
$cos ⁡ open parenthesis 20 degrees close parenthesis cos ⁡ open parenthesis 50 degrees close parenthesis minus sin ⁡ open parenthesis 20 degrees close parenthesis sin ⁡ open parenthesis 50 degrees close parenthesis$

$cos ⁡ open parenthesis 2 times 24 degrees close parenthesis$
$2 cos ⁡ open parenthesis 24 degrees close parenthesis$
$2 cos squared ⁡ open parenthesis 24 degrees close parenthesis minus 1$

$0.6691$
$1.8271$
$0.6691$

$cos ⁡ open parenthesis 3 times 49 degrees close parenthesis$
$3 cos ⁡ open parenthesis 49 degrees close parenthesis$
$4 cos cubed ⁡ open parenthesis 49 degrees close parenthesis minus 3 cos ⁡ open parenthesis 49 degrees close parenthesis$

$cos squared ⁡ open parenthesis 17 degrees close parenthesis plus sin squared ⁡ open parenthesis 17 degrees close parenthesis$
$cos squared ⁡ open parenthesis 86 degrees close parenthesis plus sin squared ⁡ open parenthesis 86 degrees close parenthesis$
$cos squared ⁡ open parenthesis 111 degrees close parenthesis plus sin squared ⁡ open parenthesis 111 degrees close parenthesis$

$\frac{1}{\cos^{2} (25^{\circ})} - \tan^{2} (25^{\circ})$
$\frac{1}{\cos^{2} (100^{\circ})} - \tan^{2} (100^{\circ})$
$\frac{1}{\cos^{2} (8^{\circ})} - \tan^{2} (8^{\circ})$

For Problems 23–28, combine like terms.

$3 x^{2} - x - 5 x^{2}$
$3 \cos^{2} (θ) - \cos (θ) - 5 \cos^{2} (θ)$

$- 2 x^{2} - x$
$- 2 \cos^{2} (θ) - \cos (θ)$

$4 x + 5 y - y$
$4 \cos (θ) + 5 \sin (θ) - \sin (θ)$

$- 3 S C + 7 S C$
$- 3 \sin (θ) \cos (θ) + 7 \sin (θ) \cos (θ)$

$4 S C$
$4 sin ⁡ open parenthesis θ close parenthesis cos ⁡ open parenthesis θ close parenthesis$

$4 S C + 11 S C - 17 S C$
$4 \sin (θ) \cos (θ) + 11 \sin (θ) \cos (θ) - 17 \sin (θ) \cos (θ)$

$- C^{2} S^{3} + 6 C^{2} S^{3}$
$- \cos^{2} (θ) \sin^{3} (θ) + 6 \cos^{2} (θ) \sin^{3} (θ)$

$5 C squared S cubed$
$5 cos squared ⁡ open parenthesis θ close parenthesis sin cubed ⁡ open parenthesis θ close parenthesis$

$7 C squared S squared minus open parenthesis 2 C S close parenthesis squared$
$7 cos squared ⁡ open parenthesis θ close parenthesis sin squared ⁡ open parenthesis θ close parenthesis minus open parenthesis 2 cos ⁡ open parenthesis θ close parenthesis sin ⁡ open parenthesis θ close parenthesis close parenthesis squared$

For Problems 29–34, simplify the expression, then evaluate.

$\cos (t) + 2 \cos (t) \sin (t) - 3 \cos (t),$ for $t equals 143 degrees$

$- 2 \cos (t) + 2 \cos (t) \sin (t); 0.6360$

$11 \tan (w) - 4 \tan (w) + 6 \tan (w) \cos (w),$ for $w equals 8 degrees$

$7 \tan (θ) - 4 \tan (ϕ) + 3 \tan (ϕ) - 6 \tan (θ),$ for $θ = 21^{\circ}, ϕ = 89^{\circ}$

$\tan (θ) - \tan (ϕ); - 56.91$

$5 \sin (A) + \sin (B) - 6 \sin (B) + 6 \sin (A),$ for $A = 111^{\circ}, B = 26^{\circ}$

$- \sin (x) \cos (x) - \sin (2 x) + 3 \sin (x) \cos (x),$ for $x equals 107 degrees$

$2 sin ⁡ open parenthesis x close parenthesis cos ⁡ open parenthesis x close parenthesis minus 2 sin ⁡ open parenthesis 2 x close parenthesis; 0$

$4 \cos (u) \sin (u) + 3 \sin (2 u) - 10 \sin (u) \cos (u),$ for $u equals 2 degrees$

For Problems 35–44, decide whether or not the expressions are equivalent. Explain.

$\cos (x + y)$ , $\cos (x) + \cos (y)$

$\sin (θ - ϕ)$ , $\sin (θ) - \sin (ϕ)$

$tan ⁡ open parenthesis 2 A close parenthesis$ , $2 tan ⁡ open parenthesis A close parenthesis$

$\cos (\frac{1}{2} β)$ , $\frac{1}{2} \cos (β)$

$(\sin α)^{2}$ , $\sin^{2} (α)$

Yes

$(\tan (B))^{2}$ , $\tan (B^{2})$

$sin ⁡ open parenthesis 3 t close parenthesis plus sin ⁡ open parenthesis 5 t close parenthesis$ , $sin ⁡ open parenthesis 8 t close parenthesis$

$3 cos ⁡ open parenthesis 2 x close parenthesis plus 5 cos ⁡ open parenthesis 2 x close parenthesis$ , $8 cos ⁡ open parenthesis 2 x close parenthesis$

$\tan (θ + 45^{\circ}) - \tan (θ)$ , $tan ⁡ open parenthesis 45 degrees close parenthesis$

$\sin (30^{\circ} + z) + \sin (30^{\circ} - z)$ , $sin ⁡ open parenthesis 60 degrees close parenthesis$

For Problems 45–56, multiply or expand.

$x open parenthesis 2 x minus 1 close parenthesis$
$sin ⁡ open parenthesis A close parenthesis open parenthesis 2 sin ⁡ open parenthesis A close parenthesis minus 1 close parenthesis$

$2 x squared minus x$
$2 sin squared ⁡ open parenthesis A close parenthesis minus sin ⁡ open parenthesis A close parenthesis$

$q open parenthesis 5 r minus q close parenthesis$
$cos ⁡ open parenthesis θ close parenthesis open parenthesis 5 sin ⁡ open parenthesis θ close parenthesis minus cos ⁡ open parenthesis θ close parenthesis close parenthesis$

$a open parenthesis b minus 3 a close parenthesis$
$tan ⁡ open parenthesis A close parenthesis open parenthesis tan ⁡ open parenthesis B close parenthesis minus 3 tan ⁡ open parenthesis A close parenthesis close parenthesis$

$a b minus 3 a squared$
$tan ⁡ open parenthesis A close parenthesis tan ⁡ open parenthesis B close parenthesis minus 3 tan squared ⁡ open parenthesis A close parenthesis$

$3 w open parenthesis 2 w minus z close parenthesis$
$3 sin ⁡ open parenthesis α close parenthesis open parenthesis 2 sin ⁡ open parenthesis α close parenthesis minus sin ⁡ open parenthesis β close parenthesis close parenthesis$

$(C + 1) (2 C - 1)$
$(\cos (ϕ) + 1) (2 \cos (ϕ) - 1)$

$2 C^{2} + C - 1$
$2 \cos^{2} (ϕ) + \cos (ϕ) - 1$

$(3 S - 2) (S + 1)$
$(3 \sin (B) - 2) (\sin (B) + 1)$

$(a + b) (a - b)$
$(\cos (θ) + \cos (ϕ)) (\cos (θ) - \cos (ϕ))$

$a^{2} - b^{2}$
$\cos^{2} (θ) - \cos^{2} (ϕ)$

$(t - w) (t + 4 w)$
$(\tan (α) - \tan (β)) (\tan (α) + 4 \tan (β))$

$open parenthesis 1 minus T close parenthesis squared$
$open parenthesis 1 minus tan ⁡ open parenthesis θ close parenthesis close parenthesis squared$

$1 - 2 T + T^{2}$
$1 - 2 \tan (θ) + \tan^{2} (θ)$

$open parenthesis 2 plus 3 S close parenthesis squared$
$open parenthesis 2 plus 3 sin ⁡ open parenthesis θ close parenthesis close parenthesis squared$

$(T^{2} + 2) (T^{2} - 2)$
$(\tan^{2} (θ) + 2) (\tan^{2} (θ) - 2)$

$T^{4} - 4$
$\tan^{4} (θ) - 4$

$(2 c^{2} - 3) (2 c^{2} + 3)$
$(2 \cos^{2} (ϕ) - 3) (2 \cos^{2} (ϕ) + 3)$

For Problems 57–70, factor.

$9 m plus 15 n$
$9 cos ⁡ open parenthesis α close parenthesis plus 15 cos ⁡ open parenthesis β close parenthesis$

$3 open parenthesis 3 m plus 5 n close parenthesis$
$3 open parenthesis 3 cos ⁡ open parenthesis α close parenthesis plus 5 cos ⁡ open parenthesis β close parenthesis close parenthesis$

$12 p minus 20 q$
$12 sin ⁡ open parenthesis θ close parenthesis minus 20 sin ⁡ open parenthesis ϕ close parenthesis$

$5 r squared minus 10 q r$
$5 tan squared ⁡ open parenthesis C close parenthesis minus 10 tan ⁡ open parenthesis B close parenthesis tan ⁡ open parenthesis C close parenthesis$

$5 r open parenthesis r minus 2 q close parenthesis$
$5 tan ⁡ open parenthesis C close parenthesis open parenthesis tan ⁡ open parenthesis C close parenthesis minus 2 tan ⁡ open parenthesis B close parenthesis close parenthesis$

$2 x squared minus 6 x y$
$2 sin squared ⁡ open parenthesis A close parenthesis minus 6 sin ⁡ open parenthesis A close parenthesis cos ⁡ open parenthesis A close parenthesis$

$9 C squared minus 1$
$9 cos squared ⁡ open parenthesis β close parenthesis minus 1$

$(3 C + 1) (3 C - 1)$
$(3 \cos (β) + 1) (3 \cos (β) - 1)$

$25 S squared minus 16 S$
$25 sin squared ⁡ open parenthesis β close parenthesis minus 16 sin ⁡ open parenthesis β close parenthesis$

$6 T cubed minus 8 T squared$
$6 tan cubed ⁡ open parenthesis A close parenthesis minus 8 tan squared ⁡ open parenthesis A close parenthesis$

$2 T squared open parenthesis 3 T minus 4 close parenthesis$
$2 tan squared ⁡ open parenthesis A close parenthesis open parenthesis 3 tan ⁡ open parenthesis A close parenthesis minus 4 close parenthesis$

$9 T squared minus 15 T cubed$
$9 tan squared ⁡ open parenthesis B close parenthesis minus 15 tan cubed ⁡ open parenthesis B close parenthesis$

$t^{2} - t - 20$
$\tan^{2} (θ) - \tan (θ) - 20$

$(t - 5) (t + 4)$
$(\tan (θ) - 5) (\tan (θ) + 4)$

$s^{2} + 5 s - 6$
$\sin^{2} (A) + 5 \sin (A) - 6$

$3 c^{2} + 2 c - 1$
$3 \cos^{2} (B) + 2 \cos (B) - 1$

$(3 c - 1) (c + 1)$
$(3 \cos (B) - 1) (\cos (B) + 1)$

$8 S^{2} - 6 S + 1$
$8 \sin^{2} (ϕ) - 6 \sin (ϕ) + 1$

$6 S^{2} - 5 S - 1$
$6 \sin^{2} (α) - 5 \sin (α) - 1$

$(6 S + 1) (S - 1)$
$(6 \sin (α) + 1) (\sin (α) - 1)$

$T^{2} - 4 T - 12$
$\tan^{2} (α) - 4 \tan (α) - 12$

Trigonometry by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.