Finding an Angle’s Sine Given the Sine of Its Complement

This post about finding an angle’s sine given the sine of its complement is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.

Question

Triangle ABC is a right triangle. Sin B = 3/8. What is the value of sin A?


Solution

“Sin” is an abbreviation for sine. The sine of angle A in a right triangle means the ratio of the length of the side opposite angle A to the length of the hypotenuse (longest side).

You are asked to find an angle’s sine given the sine of its complement. A diagram should help. We can make some guesses about how triangle ABC looks. It’s conventional to name the right angle of a right triangle C, so right triangle ABC may look like this:

Right triangle ABC. Side c is the hypotenuse, the longest side.

Right triangle ABC

Right triangle ABC. Side AB, the hypotenuse, is labelled 8 units long and side AC is labelled 3 units long.

Right triangle ABC.

That the sine of angle B is 3/8 means the length of the side opposite angle divided by the length of the hypotenuse is 3/8. 

Those lengths may be 3 and 8 (or 30 and 80 or 0.3 and 0.8 or any combination in the proportion of 3/8).

The sine of angle is equal to the length of the side opposite angle A divided by the length of the hypotenuse. What is the length of the side opposite angle A?

Sidebar: Pythagorean Theorem.

The Pythagorean Theorem states that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the legs.

Right triangle ABC. The side opposite angle A is labelled a; the side opposite angle B is labelled b; the side opposite angle C is labelled c.

Right triangle ABC

 In other words, using the labels in the triangle at left, where each side is named for the angle opposite it, .

Back to the Question

We need to know a. Solve the Pythagorean equation above for a:

And we know that and . Then

Now the triangle looks like this:

Right triangle ABC. The side opposite angle B is labelled 3; the side opposite angle C is labelled 8; and the side opposite angle A is labelled "square root of 55."

Right triangle ABC.

The sine of A is the length of side opposite A divided by the length of the hypotenuse:

Read more about sine.

This question is similar to question number 20 in the sample questions for the Accuplacer Advanced Algebra and Functions test.

How Do You Evaluate a Function? II

\fn_jvn {\color{Blue} f(x)=x^2-2x+4}This post about how to evaluate a function is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.

Question

For the function

\fn_jvn {\color{Blue} f(x)=x^2-2x+4}

what is the value of \inline \fn_jvn f(x-1)?

A.  \inline \fn_jvn x^2-2x+3

B. \inline \fn_jvn (x-1)^2-2(x-1)+4\inline \fn_jvn (x-1)(x^2-2x+3)

C. \inline \fn_jvn (x-1)(x^2-2x+3)

D.  \fn_jvn x^2+3x+4


Solution

f(x-1) is what you get when you replace x with x-1 in . That is called “evaluating the function for x-1.

Sidebar: Function Notation

In algebra, usually when you write two things side by side with one of them in parentheses, that means you multiply those two things together:

\fn_jvn {\color{Blue} a(b)=a \cdot b}

So no one can blame you for thinking f(x) means f multiplied by x. But math notation, like other kinds of language, has exceptions. f(x) means function f, carried out on x. And f(something else) means function f carried out on something else. For example, say function g is defined as . That means ; ; and means  (whatever * means).

It gets interesting when the parentheses in the function definition contain a function. For example, if  , what is ? Well, we said that if you want to find , you insert * into the function definition. To find , insert  into the function definition: if , then .

So to find f(x-1) in the expression

\fn_jvn {\color{Blue} f(x)=x^2-2x+4}

insert x-1 wherever you see x. You should get:

\fn_jvn {\color{Blue} (x-1)^2-2(x-1)+4}

That’s answer choice B.

Usually an answer like this would be expressed as , but this question is multiple choice so you go with what you’ve got.

Another example of how to evaluate a function is here.

This question is similar to question number 13 in the sample questions for the Accuplacer Advanced Algebra and Functions test.

Congruent Triangles

This post about congruent triangles is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.

Question

Angle A in triangle ABC is congruent to angle D in triangle DEF. Which of the statements below, if it’s true, proves that triangles ABC and DEF are congruent?

A.  and

B.   and

C.   and

D.   and

Triangle DEF. Angle D is marked congruent to angle A in triangle ABC, which is shown in a different picture.

Triangle DEF.

Triangle ABC. Angle A is marked congruent to angle D in triangle DEF, which is shown in a different picture.

Triangle ABC.


Solution

Two triangles are congruent if one of these things is true:

  • every side of one is congruent to a side of the other (SSS);
  • two sides and the angle between them in one are congruent to two sides and the angle between them in the other (SAS); or
  • two angles and any side in one are congruent to two angles and any side of the other (ASA or AAS).

Let’s go through the answer choices to see if one of them meets one of those criteria.

Triangle DEF. Angle D is marked congruent to angle A in triangle ABC, angle F is marked congruent to angle C, and side DF is marked congruent to side AC. Triangle ABC is shown in a different picture.

Triangle DEF

A.  and .

Triangle ABC. Angle A is marked congruent to angle D in triangle DEF, angle C is marked congruent to angle F, and side AC is marked congruent to side DF. Triangle DEF is shown in a different picture.

Triangle ABC

That’s  two  angles and the side between them in one triangle being congruent to the corresponding parts in the other triangle. That’s enough to meet the first criterion above, so it looks like that’s the answer. But let’s check the others.

 

 

 

 

B.   and  

Triangle ABC. Angle A is marked congruent to angle D in triangle DEF, side BC is marked congruent to side EF, and side AC is marked congruent to side DF.

Triangle ABC.

That’s two sides and an angle, but not the angle between them, so it does not do the job.

Triangle DEF. Angle D is marked congruent to angle A in triangle ABC, side EF is marked congruent to side BC, and side DF is marked congruent to side AC. Triangle DEF is shown in a different picture.

Triangle DEF

 

 

 

 

 

 

 

 

 

C.   and .  Same problem as answer choice B: Two sides and one angle of one are congruent to two sides and one angle of the other, but the angle is not the one between the two sides, so that information is not enough for a proof.

Triangle DEF. Angle D is marked congruent to angle A in triangle ABC, side EF is marked congruent to side BC, and side DE is marked congruent to side AB. Triangle ABC is shown in a different picture.

Triangle DEF

Triangle ABC. Angle A is marked congruent to angle D in triangle DEF, side BC is marked congruent to side EF, and side AB is marked congruent to side DE. Triangle DEF is shown in a different picture.

Triangle ABC

 

 

 

 

 

 

D.  and . This gives you only congruent angles. Every proof of congruent triangles requires at least one side.

Triangle ABC. Angle A is marked congruent to angle D in triangle DEF, angle B is marked congruent to angle E, and angle C is marked congruent to angle F. Triangle DEF is shown in a different picture.

Triangle ABC

 

The correct answer

Triangle DEF. Angle D is marked congruent to angle A in triangle ABC, angle E is marked congruent to angle B, and angle F is marked congruent to angle C. Triangle ABC is shown in a different picture.

Triangle DEF

is choice A.

This question is similar to question number 16 in the sample questions for the Accuplacer Advanced Algebra and Functions test.

How Do You Evaluate a Function? I

This post about how you evaluate a function is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.

To evaluate a function means to find the function’s y-value if you are given an x-value.

Question

Let’s say you’re given this function:

and you’re asked to find f(10) — in other words, the value of f where x=10.

Solution

To do that, substitute 10 into the equation for f where you see x:

Now clean it up: Substitute 8 for 10-2:

And multiply 4 by 8:

\fn_jvn {\color{Blue} f(10)=32}

So f(10)=32..

And that’s all there is to evaluating this function.

The example in this post is a linear function.

See How Do You Evaluate a Function? II for an example of how to evaluate a quadratic function.

This question is similar to question number 1 in the sample questions for the Accuplacer Advanced Algebra and Functions test.