Trig Algebra with Trigonometric Ratios

Section 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.

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 \(-16t^2 + 64t\text{,}\) 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
\begin{equation*} -16(\alert{1})^2 + 64(1\alert{1}) = -16 + 64 = 48~ \text{feet} \end{equation*}
and after 2 seconds, the height is
\begin{equation*} -16(\alert{2})^2 + 64(\alert{2}) = -64 + 128 = 64~ \text{feet} \end{equation*}
and so on.

Subsection 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.

Note 5.1.

Keep in mind that \(\sin (X)\text{,}\) for example, is not a product: it does not mean "sin times X" (whatever that might be). Instead, "sin" is the name of a function, and \(\sin (X)\) means to evaluate the sine function at X. Thus, \(\sin (X)\) is a single number, the output of the sine function.

Example 5.2.

Evaluate each expression for \(X = 30\degree\) and \(Y = 135\degree\text{.}\)

\(\displaystyle 2 \tan (Y) + 3\sin (X)\)
\(\displaystyle 6\tan (X) \cos (Y)\)

Solution.

Substituting the values for \(X\) and \(Y\text{,}\) we get
\begin{equation*} 2 \tan (\alert{135\degree}) + 3\sin (\alert{30\degree}) \end{equation*}
Next, we evaluate each trig ratio and follow the order of operations.
\begin{align*} 2 \tan (\alert{135\degree}) + 3\sin (\alert{30\degree}) \amp = 2(-1) + 3(\dfrac{1}{2})\\ \amp= -2 + \dfrac{3}{2} =\dfrac{-1}{2} \end{align*}
This expression includes the product of two trig ratios, \(\tan (X) \cos (Y)\text{.}\)
\begin{align*} 6\tan (X) \cos (Y)\amp = 6\tan (\alert{30\degree}) \cos (\alert{135\degree})\\ \amp= 6\left(\frac{1}{\sqrt{3}}\right) \left(\dfrac{-1}{\sqrt{2}}\right) \\ \amp =\dfrac{-6}{\sqrt{6}} = -\sqrt{6} \end{align*}

Checkpoint 5.3.

Evaluate each expression for \(X = 30\degree\text{,}\) \(Y = 60\degree\text{.}\)

\(\displaystyle 4\sin (3X + 45\degree)\)
\(\displaystyle 1 - \cos(4Y)\)

Answer.

\(\displaystyle 2\sqrt{2}\)
\(\displaystyle \dfrac{3}{2}\)

Caution 5.4.

In the previous Exercise, \(\sin (3X + 45\degree)\) is not equal to \(\sin (3X) + \sin (45\degree)\text{.}\) That is,

\begin{equation*} \sin (90\degree + 45\degree) \not= \sin (90\degree) + \sin (45\degree) \end{equation*}

(You can check this for yourself.) We must follow the order of operations and evaluate the expression \(3X + 45\degree\) inside parentheses before applying the sine function.

Subsection Simplifying Trigonometric Expressions

Because expressions such as \(\sin (x)\) and \(\tan (\theta)\) 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.

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{align*} 2x(x - 6) + 3(x + 2) \amp = 2x^2 - 12x + 3x + 6\\ \amp = 2x^2 - 9x + 6 \end{align*}

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 \(x\text{.}\) For instance, you can check that, at \(x = \alert{3}\text{,}\)

\begin{align*} 2(\alert{3})(\alert{3} - 6) + 3(\alert{3} + 2)\amp = 6(-3) + 3(5) = -3\\ \text{and}~~~~~~~~~~~~~~~ 2(\alert{3})^2 - 9(\alert{3}) + 6 \amp = 18 - 27 + 6 = -3 \end{align*}

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

\begin{alignat*}{4} \amp 8xy\amp \amp -{}\amp\amp 6xy \amp\amp = 2xy\\ \amp 8\cos (\theta) \sin (\theta)\amp \amp -{}\amp \amp 6\cos(\theta) \sin (\theta) \amp \amp = 2\cos (\theta) \sin (\theta) \end{alignat*}

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

Example 5.5.

Simplify.

\(\displaystyle 3\tan (A) + 4\tan (A) - 2\cos (A)\)
\(\displaystyle 2 - \sin (B) + 2\sin (B)\)

Solution.

Combine like terms.

\begin{equation*} 3\tan (A) + 4\tan (A) - 2\cos (A) = 7 \tan (A) - 2\cos (A) \end{equation*}

Note that \(\tan (A)\) and \(\cos (A)\) are not like terms.
Combine like terms.

\begin{equation*} 2 - \sin (B) + 2\sin (B) = 2 + \sin (B) \end{equation*}

Note that \(~-\sin (B)\) is treated as \(~-1\cdot \sin (B)\text{.}\)

Checkpoint 5.6.

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

Answer.

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

Caution 5.7.

In the previous Checkpoint, note that \(\cos (t)\) and \(\cos (w)\) are not like terms. (We can choose values for \(t\) and \(w\) so that \(\cos (t)\) and \(\cos (w)\) have different values.)

Example 5.8.

Simplify, and evaluate for \(z = 40\degree\text{.}\)

\begin{equation*} 3 \sin (z) - \sin (z) \tan (z) + 3 \sin (z) \end{equation*}

Solution.

We can combine like terms to get

\begin{equation*} 6 \sin (z) - \sin (z) \tan (z) \end{equation*}

Because \(40\degree\) is not one of the angles for which we know exact trig values, we use a calculator to evaluate the expression.

\begin{align*} 6 \sin (40\degree) - \sin (40\degree)\tan (40\degree) \amp = 6 (0.6428)(0.8391)\\ \amp = 3.3174 \end{align*}

Checkpoint 5.9.

Simplify, and evaluate for \(x = 25\degree,~y = 70\degree\)

\begin{equation*} 3 \cos (x) + \cos (y) - 2 \cos (y) + \cos (x) \end{equation*}

Answer.

\(3.2832\)

Subsection Powers of Trigonometric Ratios

Compare the two expressions

\begin{equation*} \Big(\cos (\theta)\Big)^2~~\text{and}~~~ \cos (\theta^2) \end{equation*}

They are not the same.

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

For example, if \(\theta = 30\degree\text{,}\) then

\begin{align*} \Big(\cos (30\degree)\Big)^2 \amp = \left(\dfrac{\sqrt{3}}{2}\right)^2 = \dfrac{3}{4}\\ \text{but}~~~~~~~~ \cos \Big((30^2) \degree\Big) \amp =\cos (900\degree) = \cos (180\degree) = -1 \end{align*}

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

The square of cosine.

\begin{equation*} \cos^2 (\theta) ~~~~ \text{means} ~~~~~\Big(\cos (\theta)\Big)^2 \end{equation*}

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

\begin{equation*} \sin^2 (\theta) = \Big(\sin (\theta)\Big)^2 ~~~\text{and}~~~ \tan^2 (\theta) = \Big(\tan (\theta)\Big)^2 \end{equation*}

Example 5.10.

Evaluate \(\sin^2 (45\degree)\text{.}\)

Solution.

\(\sin^2 (45\degree) = \Big(\sin (45\degree)\Big)^2 = \left(\dfrac{1}{\sqrt{2}}\right)^2 = \dfrac{1}{2}\)

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

Checkpoint 5.11.

Evaluate \(\tan^4 (60\degree)\)

Answer.

\(9\)

Subsection 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

\begin{equation*} x(3x - 2) = 3x^2 - 2x \end{equation*}

and

\begin{equation*} (x - 3)(x + 5) = x^2 + 2x - 15 \end{equation*}

Example 5.12.

Using the distributive law, multiply \(~~\cos (t) \Big(3 \cos (t) - 2\Big)\text{.}\)

Solution.

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{align*} \cos (t) \Big(3\cos (t) - 2\Big)\amp = \Big(\cos (t)\Big)\Big(3\cos (t)\Big) - \Big(\cos (t)\Big)\cdot 2 \\ \amp = 3\cos^2 (t) - 2\cos (t) \\ \blert{x (3x-2)} \amp \blert{= x\cdot 3x - x\cdot 2}\\ \amp\blert{=3x^2 - 2x} \end{align*}

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

Checkpoint 5.13.

Multiply \(~~2\tan (\beta) \Big(4 \tan^2 (\beta) + \tan (\alpha)\Big)\)

Answer.

\(8 \tan^3 (\beta) + 2 \tan (\beta) \tan (\alpha)\)

We can also use the distributive law to multiply binomials that include trig ratios. You may have used the acronym \(\blert{FOIL}\) to remember the four multiplications in a product of binomials:

\(~~~~~~~~\blert{F}\)irst terms, \(\blert{O}\)utside terms, \(\blert{I}\)nside terms, and \(\blert{L}\)ast terms.

Example 5.14.

Multiply \(~~\Big(4\sin (C) - 1\Big)\Big(3 \sin (C) + 2\Big)\text{.}\)

Solution.

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

\begin{align*} (4x - 1)(3x + 2) \amp = 4x\cdot 3x + 4x\cdot 2 - 1\cdot 3x - 1\cdot 2\\ \amp = 12x^2 + 8x - 3x - 2 = 12x^2 + 5x - 2 \end{align*}

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

\begin{alignat*}{1} \Big(4\amp\sin (C) - 1\Big)\Big(\sin (C) + 2\Big)\\ \amp = \Big(4\sin (C)\Big)\Big(3\sin (C)\Big) + \Big(4\sin (C)\Big)\cdot 2 - 1\Big(3\sin (C)\Big) - 1\cdot 2\\ \amp = 12\sin^2 (C) + 5\sin (C) - 2 \end{alignat*}

Checkpoint 5.15.

Expand \(~~\Big(4\cos (\alpha) + 3\Big)^2\)

Answer.

\(16 \cos^2 (\alpha) + 24 \cos (\alpha) + 9\)

Subsection 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.

Example 5.16.

Factor.

\(\displaystyle 6w^2 - 9w\)
\(\displaystyle 6 \sin^2 (\theta) - 9\sin (\theta)\)

Solution.

We factor out the common factor, \(3w\text{.}\)

\begin{equation*} 6w^2 - 9w = 3w(2w-3) \end{equation*}
We factor out the common factor, \(3\sin (\theta)\text{.}\)

\begin{equation*} 6\sin^2 (\theta)- 9\sin (\theta) = 3\sin (\theta)\Big(2\sin (\theta)-3\Big) \end{equation*}

Checkpoint 5.17.

Factor.

\(\displaystyle 2a^2 - ab\)
\(\displaystyle 2 \cos^2 (\phi) - \cos (\phi) \sin (\phi)\)

Answer.

\(\displaystyle a(2a - b)\)
\(\displaystyle \cos (\phi) \Big(2\cos (\phi) - \sin (\phi)\Big)\)

We can also factor quadratic trinomials.

Example 5.18.

Factor.

\(\displaystyle t^2 - 3t - 10\)
\(\displaystyle \tan^2 (\alpha) - 3\tan (\alpha) - 10\)

Solution.

We look for numbers \(p\) and \(q\) so that \((t + p)(t + q) = t^2 - 3t - 10\text{.}\) We see that their product must be \(~pq = -10~\text{,}\) and their sum must be \(~p + q = -3\text{.}\) By checking the factors of \(-10\) for the correct sum, we find \(p = -5\) and \(q = 2\text{.}\) Thus,

\begin{equation*} t^2 - 3t - 10 = (t - 5)(t + 2) \end{equation*}
Now replace \(t\) by \(\tan (\alpha)\) to find

\begin{equation*} \tan^2 (\alpha) - 3\tan (\alpha) - 10 = \Big(\tan (\alpha) - 5\Big)\Big(\tan (\alpha) + 2\Big) \end{equation*}

Checkpoint 5.19.

Factor.

\(\displaystyle 3z^2 - 2z - 1\)
\(\displaystyle 3 \sin^2 (\beta) - 2\sin (\beta) - 1\)

Answer.

\(\displaystyle (3z + 1)(z - 1)\)
\(\displaystyle \Big(3\sin (\beta) + 1\Big)\Big(\sin (\beta) - 1\Big)\)

Review the following skills you will need for this section.

Algebra Refresher 5.1.

Factor.

\(\displaystyle 2x^2 + 5x - 12\)
\(\displaystyle 3x^2 - 2x - 8\)
\(\displaystyle 12x^2 + x - 1\)
\(\displaystyle 6x^2 - 13x - 15\)
\(\displaystyle 6x^2 - x - 12\)
\(\displaystyle 24x^2 + 10x - 21\)

\(\underline{\qquad\qquad\qquad\qquad}\)

Algebra Refresher Answers

\(\displaystyle (2x - 3)(x + 4)\)
\(\displaystyle (3x + 4)(x - 2)\)
\(\displaystyle (4x - 1)(3x + 1)\)
\(\displaystyle (6x + 5)(x - 3)\)
\(\displaystyle (2x - 3)(3x + 4)\)
\(\displaystyle (6x + 7)(4x - 3)\)

Subsection Section 5.1 Summary

Subsubsection Vocabulary

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

Subsubsection 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 \(\cos^2 (\theta)\) to denote \((\cos 9)^2\text{,}\) and \(\cos^n (\theta)\) to denote \((\cos (\theta))^n\text{.}\) (Similarly for the other trig ratios.)
We can factor trigonometric expressions with the same techniques we use for algebraic expressions.

Subsubsection Study Questions

To evaluate \(\cos^2 (30\degree)\text{,}\) Delbert used the keystrokes
\begin{equation*} \boxed {\text{COS}}~~ 30~~ \boxed{x^2}~~ \boxed{\text{ENTER}} \end{equation*}
and got the answer 1. Were his keystrokes correct? Why or why not?
Make up an example to show that \(\tan (\theta + \phi) \not= \tan (\theta) + \tan (\phi)\text{.}\)
Factor each expression, if possible.
1. \(\displaystyle x^2 - 4x\)
2. \(\displaystyle x^2 - 4\)
3. \(\displaystyle x^2 + 4\)
4. \(\displaystyle x^2 + 4x + 4\)
5. \(\displaystyle x^2 - 4x + 4\)
6. \(\displaystyle x^2 + 4x\)
7. \(\displaystyle x^2 - 4x - 4\)
8. \(\displaystyle -x^2 + 4\)

Subsubsection 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

Exercises Homework 5.1

Exercise Group.

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

1.

\(5 \tan (135\degree) + 6 \cos (60\degree)\)

2.

\(3 \tan (240\degree) + 8 \sin (300\degree)\)

3.

\(\sin (15\degree + 30\degree)\)

4.

\(\cos(2 \cdot 75\degree)\)

5.

\(8 \cos^2 (30\degree)\)

6.

\(12 \sin^2 (315\degree)\)

7.

\(3 \tan^2 (150\degree) - \sin^2 (45\degree)\)

8.

\(1 + \tan^2 (120\degree)\)

Exercise Group.

For Problems 9–16, evaluate the expressions for \(x = 30\degree,~y = 45\degree\text{,}\) and \(z = 60\degree\text{.}\) Give exact values for your answers.

9.

\(3 \sin (x) + 5 \cos (y)\)

10.

\(4 \tan (y) + 6 \cos (y)\)

11.

\(-2 \tan (3y)\)

12.

\(\sin (3z - 2x)\)

13.

\(\cos^2 (x) + \sin^2 (x)\)

14.

\(7 \sin^2 (y) + 7 \cos^2 (y)\)

15.

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

16.

\(\tan (180\degree - x) \tan (x)\)

Exercise Group.

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

17.

\(\displaystyle \sin (10\degree + 40\degree)\)
\(\displaystyle \sin (10\degree) + \sin (40\degree)\)
\(\displaystyle \sin (10\degree) \cos (40\degree) + \cos (10\degree) \sin (40\degree)\)

18.

\(\displaystyle \cos (20\degree + 50\degree)\)
\(\displaystyle \cos (20\degree) + \cos (50\degree)\)
\(\displaystyle \cos (20\degree) \cos (50\degree) - \sin (20\degree) \sin (50\degree)\)

19.

\(\displaystyle \cos (2 \cdot 24\degree)\)
\(\displaystyle 2 \cos (24\degree)\)
\(\displaystyle 2 \cos^2 (24\degree) - 1\)

20.

\(\displaystyle \cos (3 \cdot 49\degree)\)
\(\displaystyle 3 \cos (49\degree)\)
\(\displaystyle 4 \cos^3 (49\degree) - 3\cos (49\degree)\)

21.

\(\displaystyle \cos^2 (17\degree) + \sin^2 (17\degree)\)
\(\displaystyle \cos^2 (86\degree) + \sin^2 (86\degree)\)
\(\displaystyle \cos^2 (111\degree) + \sin^2 (111\degree)\)

22.

\(\displaystyle \dfrac{1}{\cos^2 (25\degree)} - \tan^2 (25\degree)\)
\(\displaystyle \dfrac{1}{\cos^2 (100\degree)} - \tan^2 (100\degree)\)
\(\displaystyle \dfrac{1}{\cos^2 (8\degree)} - \tan^2 (8\degree)\)

Exercise Group.

For Problems 23–28, combine like terms.

23.

\(\displaystyle 3x^2 - x - 5x^2\)
\(\displaystyle 3\cos^2 (\theta) - \cos (\theta) - 5\cos^2 (\theta)\)

24.

\(\displaystyle 4x + 5y - y\)
\(\displaystyle 4\cos (\theta) + 5\sin (\theta) - \sin (\theta)\)

25.

\(\displaystyle -3SC + 7SC\)
\(\displaystyle -3\sin (\theta) \cos (\theta) + 7\sin (\theta) \cos (\theta)\)

26.

\(\displaystyle 4SC + 11SC - 17SC\)
\(\displaystyle 4\sin (\theta) \cos (\theta) + 11\sin (\theta) \cos (\theta) - 17\sin (\theta) \cos (\theta)\)

27.

\(\displaystyle -C^2S^3 + 6C^2S^3\)
\(\displaystyle -\cos^2 (\theta) \sin^3 (\theta) + 6 \cos^2 (\theta)\sin^3 (\theta)\)

28.

\(\displaystyle 7C^2S^2 - (2CS)^2\)
\(\displaystyle 7\cos^2 (\theta) \sin^2 (\theta) - (2\cos (\theta) \sin (\theta))^2\)

Exercise Group.

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

29.

\(\cos (t) + 2 \cos (t) \sin (t) - 3 \cos (t),\) for \(t = 143\degree\)

30.

\(11 \tan (w) - 4 \tan (w) + 6 \tan (w) \cos (w),\) for \(w = 8\degree\)

31.

\(7 \tan (\theta) - 4 \tan (\phi) + 3 \tan (\phi) - 6 \tan (\theta),\) for \(\theta = 21 \degree, ~ \phi = 89\degree\)

32.

\(5 \sin (A) + \sin (B) - 6\sin (B) + 6 \sin (A),\) for \(A = 111\degree,~ B = 26\degree\)

33.

\(-\sin (x) \cos (x) - \sin (2x) + 3\sin (x) \cos (x),\) for \(x = 107\degree\)

34.

\(4 \cos (u) \sin (u) + 3 \sin (2u) - 10 \sin (u) \cos (u),\) for \(u = 2\degree\)

Exercise Group.

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

35.

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

36.

\(\sin (\theta - \phi),~~\sin (\theta) - \sin (\phi)\)

37.

\(\tan (2A),~~ 2 \tan (A)\)

38.

\(\cos (\frac{1}{2}\beta),~~\frac{1}{2}\cos (\beta)\)

39.

\((\sin \alpha)^2,~~\sin^2 (\alpha)\)

40.

\((\tan (B))^2,~~ \tan (B^2)\)

41.

\(\sin (3t) + \sin (5t),~~\sin (8t)\)

42.

\(3\cos (2x) + 5\cos (2x),~~ 8\cos (2x)\)

43.

\(\tan (\theta + 45\degree) - \tan (\theta),~~ \tan (45\degree)\)

44.

\(\sin (30\degree + z) + \sin (30\degree - z),~~ \sin (60\degree)\)

Exercise Group.

For Problems 45–56, multiply or expand.

45.

\(\displaystyle x(2x-1)\)
\(\displaystyle \sin (A) \Big(2\sin (A) - 1\Big)\)

46.

\(\displaystyle q(5r-q)\)
\(\displaystyle \cos (\theta) \Big(5\sin (\theta) - \cos (\theta)\Big)\)

47.

\(\displaystyle a(b - 3a)\)
\(\displaystyle \tan (A) \Big(\tan (B) - 3 \tan (A)\Big)\)

48.

\(\displaystyle 3w(2w - z)\)
\(\displaystyle 3 \sin (\alpha) \Big(2 \sin (\alpha) - \sin (\beta)\Big)\)

49.

\(\displaystyle (C + 1)(2C - 1)\)
\(\displaystyle \Big(\cos (\phi) + 1\Big)\Big(2\cos (\phi) - 1\Big)\)

50.

\(\displaystyle (3S - 2)(S + 1)\)
\(\displaystyle \Big(3\sin (B) - 2\Big)\Big(\sin (B) + 1\Big)\)

51.

\(\displaystyle (a + b)(a - b)\)
\(\displaystyle \Big(\cos (\theta) + \cos (\phi)\Big)\Big(\cos (\theta) - \cos (\phi)\Big)\)

52.

\(\displaystyle (t - w)(t + 4w)\)
\(\displaystyle \Big(\tan (\alpha) - \tan (\beta)\Big)\Big(\tan (\alpha) + 4 \tan (\beta)\Big)\)

53.

\(\displaystyle (1 - T)^2\)
\(\displaystyle \Big(1 - \tan (\theta)\Big)^2\)

54.

\(\displaystyle (2 + 3S)^2\)
\(\displaystyle \Big(2 + 3\sin (\theta)\Big)^2\)

55.

\(\displaystyle (T^2 + 2)(T^2 - 2)\)
\(\displaystyle \Big(\tan^2 (\theta) + 2\Big)\Big(\tan^2 (\theta) - 2\Big)\)

56.

\(\displaystyle (2c^2 - 3)(2c^2 + 3)\)
\(\displaystyle \Big(2\cos^2 (\phi) - 3\Big)\Big(2\cos^2 (\phi) + 3\Big)\)

Exercise Group.

For Problems 57–70, factor.

57.

\(\displaystyle 9m + 15n\)
\(\displaystyle 9\cos(\alpha) + 15\cos(\beta)\)

58.

\(\displaystyle 12p - 20q\)
\(\displaystyle 12\sin (\theta) - 20 \sin (\phi)\)

59.

\(\displaystyle 5r^2 - 10 qr\)
\(\displaystyle 5\tan^2 (C) - 10 \tan (B) \tan (C)\)

60.

\(\displaystyle 2x^2 - 6xy\)
\(\displaystyle 2\sin^2 (A) - 6\sin (A) \cos (A)\)

61.

\(\displaystyle 9C^2 - 1\)
\(\displaystyle 9\cos^2 (\beta) - 1\)

62.

\(\displaystyle 25S^2 - 16S\)
\(\displaystyle 25\sin^2 (\beta) - 16 \sin (\beta)\)

63.

\(\displaystyle 6T^3 - 8T^2\)
\(\displaystyle 6\tan^3 (A) - 8 \tan^2 (A)\)

64.

\(\displaystyle 9T^2 - 15T^3\)
\(\displaystyle 9\tan^2 (B) - 15\tan^3 (B)\)

65.

\(\displaystyle t^2 - t - 20\)
\(\displaystyle \tan^2 (\theta) - \tan (\theta) - 20\)

66.

\(\displaystyle s^2 + 5s - 6\)
\(\displaystyle \sin^2 (A) + 5 \sin (A) - 6\)

67.

\(\displaystyle 3c^2 + 2c - 1\)
\(\displaystyle 3\cos^2 (B) + 2\cos (B) - 1\)

68.

\(\displaystyle 8S^2 - 6S + 1\)
\(\displaystyle 8\sin^2 (\phi) - 6 \sin (\phi) + 1\)

69.

\(\displaystyle 6S^2 - 5S - 1\)
\(\displaystyle 6\sin^2 (\alpha) - 5\sin (\alpha) - 1\)

70.

\(\displaystyle T^2 - 4T - 12\)
\(\displaystyle \tan^2 (\alpha) - 4\tan (\alpha) - 12\)

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