The number of radians in a 720-degree angle can be written as a\pi, where a is a constant. What is the value of a?

C. 4. There are $\pi$ radians in a $180^\circ$ angle. An angle measure of $720^\circ$ is 4 times greater than an angle measure of $180^\circ$. Therefore, the number of radians in a $720^\circ$ angle is $4\pi$.

In a right triangle, the tangent of one of the two acute angles is \frac{\sqrt{3}}{3}. What is the tangent of the other acute angle?

D. \frac{3}{\sqrt{3}}. The tangent of a nonright angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. Using that definition for tangent, in a right triangle with legs that have lengths $a$ and $b$, the tangent of one acute angle is $\frac{a}{b}$ and the tangent for the other acute angle is $\frac{b}{a}$. It follows that the tangents of the acute angles in a right triangle are reciprocals of each other. Therefore, the tangent of the other acute angle in the given triangle is the reciprocal of $\frac{\sqrt{3}}{3}$ or $\frac{3}{\sqrt{3}}$.

SAT Math skill page

SAT Trigonometry Practice

Use side ratios and special-triangle patterns to connect angles with missing lengths.

Practice trigonometry in Dolphin SAT Practice questions

12-18 min practice time3 questions on pageGeometry and Trigonometry

Practice time12-18 min

Question bank3 questions

Best forGeometry and Trigonometry

What this tests

What to know for this SAT skill

Choose sine, cosine, or tangent from the given sides
Use trigonometric ratios to find a missing length
Convert angle measures between degrees and radians
Recognize 30-60-90 and 45-45-90 triangle ratios
Keep opposite and adjacent sides tied to the marked angle

Dolphin question bank

Practice your skill

Question 1 of 3Medium

The number of radians in a 720-degree angle can be written as

aπ

, where a is a constant. What is the value of a?

Show answer and solution

Correct answerC. 4

How to solve it

Plan: Convert the angle to radians, then match the coefficient of $π$ .

Use the conversion $18 0^{\circ} = π$ radians.
Multiply by the unit conversion: $72 0^{\circ} \times \frac{π}{18 0 ^{\circ}} = 4 π$ .
The result has the form $aπ$ , so $a = 4$ .

CheckA $72 0^{\circ}$ angle is two full rotations, and two rotations equal $4 π$ radians.

Common trapDo not divide by $π$ . Convert degrees to radians by multiplying by $\frac{π}{180}$ .

Question 2 of 3Hard

In a right triangle, the tangent of one of the two acute angles is

\frac{3}{3}

. What is the tangent of the other acute angle?

$\frac{3}{3}$
$\frac{1}{3}$
3
$\frac{3}{3}$

Show answer and solution

Correct answer $D . \frac{3}{3}$

How to solve it

Plan: Use the fact that the two acute angles in a right triangle are complementary.

For one acute angle, $tan θ = \frac{opposite}{adjacent} = \frac{3}{3}$ .
For the other acute angle, the opposite and adjacent legs switch, so its tangent is the reciprocal.
The reciprocal is $\frac{3}{3} = 3$ , which is choice D.

CheckBoth acute angles have positive tangent values, and $\frac{3}{3} \cdot 3 = 1$ .

Common trapThe second angle does not have the same tangent; complementary acute angles swap the opposite and adjacent legs.

Question 3 of 3Hard

Triangle ABC is similar to triangle DEF, where angle A corresponds to angle D and angles C and F are right angles. The length of

\overline{A B}

is 2.9 times the length of

\overline{D E}

. If

tan A = \frac{21}{20}

, what is the value of

sin D

0.724
0.690
1.05
0.345

Show answer and solution

Correct answerA. 0.724

How to solve it

Plan: Turn the tangent ratio into side lengths, then use similarity to transfer the angle ratio.

Because $tan A = \frac{21}{20}$ , represent the legs opposite and adjacent to $A$ as $21 k$ and $20 k$ .
The hypotenuse is $(21 k)^{2} + (20 k)^{2} = 29 k$ .
Thus $sin A = \frac{21 k}{29 k} = \frac{21}{29} \approx 0.724$ .
Since $A$ corresponds to $D$ , the triangles are similar and $sin D = sin A \approx 0.724$ .

CheckA sine value must lie between 0 and 1, so $0.724$ is reasonable.

Common trapThe 2.9 scale factor changes side lengths, not trigonometric ratios of corresponding angles.

Avoid these traps

Common mistakes on this skill

Using the wrong side ratio

Label opposite, adjacent, and hypotenuse relative to the marked angle before choosing a trig function.

Changing the reference angle

Opposite and adjacent switch when the reference angle changes, even though the triangle stays the same.

Mixing up special-triangle ratios

Write the 30-60-90 or 45-45-90 ratio before scaling it to the given side.

Treating degrees and radians as interchangeable

Use 180 degrees = pi radians, then multiply by pi/180 to convert a degree measure to radians.

Study plan

How to practice this skill in Dolphin

Mark the right angle, hypotenuse, and reference angle.
Label the opposite and adjacent legs relative to that angle.
Choose the ratio that uses the known side and requested side.
If the angle uses degrees or radians, convert with 180 degrees = pi radians before solving.
Estimate the result to confirm the side length is reasonable.

Practice trigonometry in Dolphin SAT

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FAQ

Questions about SAT Trigonometry Practice

Which trigonometry functions are tested on the SAT?

SAT questions can use sine, cosine, and tangent, along with special right-triangle relationships.

Do I need to memorize SOH-CAH-TOA?

It is a useful shortcut for remembering the three side ratios, even though the SAT provides some geometry formulas.

How do I convert degrees to radians?

Multiply the degree measure by pi/180. To convert radians to degrees, multiply by 180/pi.

How is this different from the right-triangles page?

The right-triangles page focuses on side lengths and the Pythagorean theorem. This page focuses specifically on angle-based trig ratios.