Right-Triangle Trigonometry: Sine, Cosine, and Tangent

Similar Right Triangles and Why Ratios Stay the Same

Right-triangle trigonometry is built on a simple idea: if two right triangles share the same acute angle, then the triangles are similar, meaning their corresponding sides are in the same proportion. Even if one triangle is “bigger,” the ratios of matching sides stay constant. Those constant ratios are what we call sine, cosine, and tangent.

To use trig ratios correctly, you must label sides relative to a chosen acute angle (not relative to the triangle itself). Pick an acute angle (call it θ), then label:

• Hypotenuse (hyp): the side opposite the right angle (always the longest side).
• Opposite (opp): the side across from θ.
• Adjacent (adj): the side next to θ that is not the hypotenuse.

Diagram-based checkpoint: label before you calculate

Imagine a right triangle with a right angle at C, and angle θ at A. The side AB is across from the right angle, so AB is the hypotenuse. The side BC is across from angle A, so BC is opposite. The side AC touches angle A and is not the hypotenuse, so AC is adjacent.

Checkpoint questions (answer mentally before moving on):

• If θ is at A, which side is the hypotenuse?
• If θ is at A, which side is opposite?
• If θ is at A, which side is adjacent?

Sine, Cosine, and Tangent as Ratios

For a right triangle and a chosen acute angle θ:

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• sin(θ) = opp / hyp
• cos(θ) = adj / hyp
• tan(θ) = opp / adj

A quick memory aid is “SOH-CAH-TOA,” but the key skill is consistent labeling: opposite and adjacent depend on which angle you chose.

Guided example 1: compute trig ratios from side lengths

A right triangle has side lengths 5, 12, and 13, with hypotenuse 13. Let θ be the acute angle whose opposite side is 5 (so the adjacent side is 12).

Step 1: Identify sides relative to θ.

• hyp = 13
• opp = 5
• adj = 12

Step 2: Compute the ratios.

• sin(θ) = opp/hyp = 5/13
• cos(θ) = adj/hyp = 12/13
• tan(θ) = opp/adj = 5/12

Now switch angles: let φ be the other acute angle. Then the roles of opposite and adjacent swap (hyp stays 13).

• sin(φ) = 12/13
• cos(φ) = 5/13
• tan(φ) = 12/5

Diagram-based checkpoint: swapping angles

In any right triangle, when you move from one acute angle to the other:

• the hypotenuse stays the hypotenuse
• opposite and adjacent swap

Checkpoint: If tan(θ) = opp/adj = 3/4 for one acute angle, what is tan of the other acute angle? (It should be the reciprocal, 4/3.)

Reciprocal Ratios: csc, sec, cot

The reciprocal trig functions are defined as the reciprocals of sine, cosine, and tangent. They are useful when a problem naturally gives you the “flipped” ratio.

• csc(θ) = 1/sin(θ) = hyp/opp
• sec(θ) = 1/cos(θ) = hyp/adj
• cot(θ) = 1/tan(θ) = adj/opp

Guided example 2: compute reciprocal ratios

Using the same 5-12-13 triangle with θ opposite 5:

• csc(θ) = hyp/opp = 13/5
• sec(θ) = hyp/adj = 13/12
• cot(θ) = adj/opp = 12/5

Notice how each reciprocal ratio uses the same three side labels; you are not learning new geometry, just new names for flipped fractions.

Finding a Missing Side (Pythagorean Theorem + Trig Ratios)

Often you will know one side and one trig ratio (or one angle and one side). In a right triangle, the Pythagorean theorem connects the side lengths:

a^2 + b^2 = c^2

where c is the hypotenuse.

Guided example 3: find a missing side, then compute trig ratios

A right triangle has legs 9 and 12. Let θ be the acute angle adjacent to the side of length 12 and opposite the side of length 9.

Step 1: Find the hypotenuse using the Pythagorean theorem.

9^2 + 12^2 = c^2  81 + 144 = c^2  225 = c^2  c = 15

Step 2: Label relative to θ.

• opp = 9
• adj = 12
• hyp = 15

Step 3: Compute trig ratios.

• sin(θ) = 9/15 = 3/5
• cos(θ) = 12/15 = 4/5
• tan(θ) = 9/12 = 3/4

Diagram-based checkpoint: label from a word description

Practice turning words into labels. Suppose a problem says: “Angle θ has an adjacent side of 7 and a hypotenuse of 25.” Before calculating anything, you should be able to state:

• adj = 7
• hyp = 25
• opp is unknown (but can be found with the Pythagorean theorem)

Then you can compute cos(θ) immediately as 7/25, and find opp to compute sin(θ) and tan(θ).

Solving for an Angle Using Inverse Trig

If you know a trig ratio value and want the angle, you use an inverse trig function. For right-triangle problems, you typically use:

• θ = sin^-1(opp/hyp)
• θ = cos^-1(adj/hyp)
• θ = tan^-1(opp/adj)

The inverse trig function returns the acute angle whose trig ratio matches the given value (in a right-triangle context).

Guided example 4: find an angle from side lengths

A right triangle has opposite side 8 and adjacent side 15 relative to angle θ.

Step 1: Choose the ratio that matches the given sides. Opposite and adjacent suggests tangent.

tan(θ) = opp/adj = 8/15

Step 2: Apply the inverse tangent.

θ = tan^-1(8/15)

Step 3: Evaluate with a calculator (in degree mode if your course is using degrees for triangle angles).

Numerically, θ ≈ tan^-1(0.5333) ≈ 28.1°.

Guided example 5: find an angle from a sine or cosine ratio

Suppose sin(θ) = 0.6 for an acute angle θ in a right triangle.

Step 1: Use the inverse sine.

θ = sin^-1(0.6)

Step 2: Evaluate.

θ ≈ 36.9°.

Diagram-based checkpoint: pick the correct inverse function

For each set of known sides relative to θ, choose the inverse trig function that uses exactly those sides:

• Known opp and hyp: use sin^-1(opp/hyp)
• Known adj and hyp: use cos^-1(adj/hyp)
• Known opp and adj: use tan^-1(opp/adj)

Short Quiz: Identify Ratios and Evaluate

Assume all angles mentioned are acute and all triangles are right triangles.

Part A: Identify the correct ratio

• 1) Which ratio equals adj/hyp? (sin, cos, tan, csc, sec, cot)
• 2) Which ratio equals hyp/opp?
• 3) Which ratio equals opp/adj?
• 4) Which ratio equals adj/opp?

Part B: Compute numeric values from a triangle

A right triangle has hypotenuse 10. Relative to angle θ, the adjacent side is 6.

• 5) Find cos(θ).
• 6) Find the opposite side length.
• 7) Find sin(θ).
• 8) Find tan(θ).

Part C: Use inverse trig to find an angle

• 9) If tan(θ) = 3/7, write an expression for θ using an inverse trig function.
• 10) If cos(θ) = 0.2, approximate θ to the nearest tenth of a degree.

Answer check (no skipping the labeling step)

For items 5–8, you should label hyp = 10, adj = 6, then compute opp using the Pythagorean theorem before evaluating sin and tan.