SAT Trig Ratios on Desmos: Sin, Cos, Tan Examples

On SAT trig ratio questions, write the ratio from the triangle first (opposite, adjacent, hypotenuse), then use Desmos only for the arithmetic. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. You measure all three from the angle the question names. When the question gives an angle in degrees, the calculator has to be in degree mode too.

You can type every line below into the College Board version of Desmos on our Desmos on the SAT page. This guide gives you the rule, three worked examples, one guided problem and two practice problems with answers. Each question is SAT-style practice, and we recomputed every answer.

The rule: name the sides from the angle

The hypotenuse sits across from the right angle. "Opposite" and "adjacent" change with the angle you stand at: the side opposite angle A is adjacent to angle B.

Two more facts handle most of the rest. The two acute angles of a right triangle add to 90°, so sin(x°) = cos(90° - x°). In words, the sine of one acute angle equals the cosine of the other. And sin²θ + cos²θ = 1, so if sin α = 3/5 in a right triangle, cos α = 4/5, from the 3-4-5 triangle.

College Board allows a calculator on the entire Math section, and Bluebook has a Desmos calculator built in with a graphing and a scientific option. Desmos will not pick the ratio for you. It takes care of the square roots, the division and the trig values of angles like 35°.

Check the angle mode before you type sin, cos or tan

A trig function of an angle gives one number in degrees and a different number in radians. Before you evaluate one, type this test line:

1  sin(30)

A result of 0.5 means the calculator is in degrees. A result of about -0.988 means it is in radians. Match the question: a degree sign (°) calls for degree mode, and π or the word "radians" calls for radian mode. Don't rely on whatever mode the calculator opened in. Run the test line whenever a question asks for a trig value of an angle.

Worked example 1 (I do): from two sides to sin A

SAT-style question. In triangle ABC, angle C is a right angle, AC = 20 and BC = 21. What is the value of sin A?

Step 1. Name the sides from angle A. BC is across from A, so BC = 21 is the opposite side. AC = 20 touches A, so it is the adjacent side. AB is across from the right angle at C, so AB is the hypotenuse, and the question doesn't give it.

Step 2. Find the hypotenuse on Desmos.

1  sqrt(20^2 + 21^2)

Line 1 returns 29, so AB = 29.

Step 3. Write the ratio. sin A = opposite/hypotenuse = 21/29. If the answer choices are decimals, type 21/29 on line 2 and compare. It is about 0.7241.

By-hand check. 20² + 21² = 400 + 441 = 841, and 29² = 841. So 20, 21 and 29 are a Pythagorean triple, and sin A = 21/29.

Notice that cos B is also 21/29, because BC is adjacent to B. That is the complementary-angle fact at work: A and B add to 90°, so sin A = cos B.

Worked example 2 (I do): from an angle to a missing side

SAT-style question. In triangle PQR, angle R is a right angle, angle P measures 35° and PR = 12. What is the length of QR, to the nearest tenth?

Step 1. Pick the ratio. Stand at P. QR is opposite P, and PR = 12 is adjacent to P. Opposite and adjacent together mean tangent.

Step 2. Set up the equation. tan(35°) = QR/12, so QR = 12 tan(35°).

Step 3. Check the mode, then compute. The question uses degrees. Type the sin(30) test line, switch to degrees if it shows -0.988, and then type:

1  12tan(35)

In degree mode, line 1 returns about 8.4025, so QR ≈ 8.4.

The mode check matters here. In radian mode, the same line reads 35 as 35 radians and returns about 5.686. You would get a wrong answer with no error message.

By-hand check. An angle smaller than 45° has a tangent smaller than 1, so the side opposite 35° must be shorter than the adjacent side of 12. 8.4 fits. 5.686 also fits that test, which is why the sin(30) line comes first.

Guided example (we do): similar right triangles

SAT-style question. Right triangles RST and XYZ are similar, with R, S and T corresponding to X, Y and Z. Angles T and Z are the right angles. If tan R = 180/299, what is the value of sin Y?

Step 1. Corresponding angles of similar triangles are equal, so angle Y equals angle S and sin Y = sin S. You can do all the work in triangle RST.

Step 2. tan R = ST/RT = 180/299. Label ST = 180 and RT = 299. Any multiple of these gives the same ratios, so these lengths are safe to use.

Step 3. Find the hypotenuse RS:

1  sqrt(180^2 + 299^2)

Line 1 returns 349.

Step 4 (your step). Stand at angle S. Which side is opposite S? Write sin S as a fraction.

Check your answer: RT = 299 is opposite S, so sin Y = sin S = 299/349. It also equals cos R, because R and S are the two acute angles of the same right triangle.

Want more questions like these? Work through the free Right-Triangle Trigonometry lesson notes.

Worked example 3: a point on the unit circle (watch the sign)

SAT-style question. Points O(0, 0), A(1, 0) and B(5/√89, -8/√89) lie in the xy-plane. Angle AOB has measure p radians. What is the value of cos(p)/sin(p)?

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Step 1. Check that B is on the unit circle. (5/√89)² + (-8/√89)² = 25/89 + 64/89 = 1, so OB has length 1.

Step 2. Decide the sign of sin(p). Angle AOB is the angle between rays OA and OB. The measure of an angle is between 0 and π radians, and sine is positive there, so sin(p) is positive even though B sits below the x-axis. The angle to B has the same size as the angle to (5/√89, 8/√89), its mirror image above the axis.

Step 3. Read the values. cos(p) = 5/√89 and sin(p) = 8/√89. The √89 cancels, so cos(p)/sin(p) = 5/8, which is 0.625.

Desmos check. Line 1 stores the angle as p. Line 2 should return 0.625:

1  p = arccos(5/sqrt(89))
2  cos(p)/sin(p)

The trap is to divide the coordinates of B directly. That gives -5/8, which describes a clockwise, directed rotation. The question asks about the angle AOB itself, so the answer is 5/8. An earlier version of this page gave -5/8. We corrected it on September 26, 2026.

To see the unit circle idea on screen, type (cos(a), sin(a)) on one line and a = π/6 on the next, in radian mode. The point lands at about (0.866, 0.5), because cos(π/6) ≈ 0.866 and sin(π/6) = 0.5. For more of these, read unit circle on Desmos.

Your turn (you do)

1. In triangle DEF, angle F is a right angle, DE = 34 and sin D = 8/17. What is the length of DF?
2. For acute angles x° and y°, sin(x°) = cos(y°). If x = 2k + 10 and y = 3k - 5, what is the value of k?

Answers.

1. 30. DE is the hypotenuse. The side opposite D is EF = 34 × 8/17 = 16. Then DF = √(34² - 16²) = √900 = 30. On Desmos, type sqrt(34^2 - 16^2).
2. k = 17. For acute angles, sin(x°) = cos(y°) means x + y = 90. So (2k + 10) + (3k - 5) = 90, 5k + 5 = 90 and k = 17. The angles are 44° and 46°.

When algebra is faster

College Board's calculator guidance says some Math questions are better done without a calculator, even though one is allowed. Many trig ratio questions fit that description. If the sides form a Pythagorean triple such as 3-4-5, 5-12-13, 8-15-17 or 20-21-29, you can write the ratio in a few seconds by hand. Cofunction questions like practice problem 2 need only x + y = 90. Save Desmos for awkward numbers such as 180 and 299, or for the trig value of an angle like 35°.

Mistakes to avoid

• Using the wrong angle mode. 12tan(35) gives about 8.40 in degrees and about 5.69 in radians. Type sin(30) first and check for 0.5.
• Naming the sides from the wrong angle. In worked example 1, 21/29 is sin A and also cos B. Before you label opposite and adjacent, find the angle the question names.
• Copying a coordinate's sign into an angle measure. In worked example 3, dividing the coordinates gives -5/8. Angle AOB has a positive sine, so the answer is 5/8.

Practice next

Start with the free Right-Triangle Trigonometry lesson notes. For more questions, browse the SAT question bank.

Related guides: for angles in radians, read unit circle on Desmos. To test a whole expression against the answer choices, see trig expressions on Desmos. For circles and perpendicular slopes, try circle tangent slope with a Desmos check.

Checked on September 26, 2026 against College Board's SAT Suite calculator policy, College Board's March 12, 2024 digital SAT launch release and the College Board version of the Desmos graphing calculator. Every answer on this page was recomputed before publishing. The Test Advantage is independent of College Board and Desmos.