Exact Values from Angle Addition Formulas

From Special Angles to New Exact Values

The exact values of sine and cosine at familiar angles such as \(30^\circ\), \(45^\circ\), and \(60^\circ\) provide more than a short list to memorize. The angle addition formulas let us combine those values to find exact values at other angles, including \(15^\circ\), \(75^\circ\), and \(105^\circ\).

The main work is organized: express the target angle as a sum or difference of angles with known values, substitute into the appropriate formula, simplify the radicals, and check the sign using the quadrant of the target angle. The angle addition formulas are established earlier; here we focus on applying them accurately.

Key idea: If \(A\) and \(B\) have known trigonometric values, the angle addition formulas can give exact values at \(A+B\) and \(A-B\). The sign of the final value must agree with the quadrant of the resulting angle.

Special-Angle Values and Addition Formulas

The basic exact sine and cosine values used below are:

\begin{array}{c|ccc} \theta & 30^\circ & 45^\circ & 60^\circ\\ \hline \sin\theta & \frac12 & \frac{\sqrt2}{2} & \frac{\sqrt3}{2}\\ \cos\theta & \frac{\sqrt3}{2} & \frac{\sqrt2}{2} & \frac12 \end{array}

For angles \(A\) and \(B\), the sine and cosine addition formulas are:

\sin(A+B)=\sin A\cos B+\cos A\sin B, \qquad \sin(A-B)=\sin A\cos B-\cos A\sin B,

\cos(A+B)=\cos A\cos B-\sin A\sin B, \qquad \cos(A-B)=\cos A\cos B+\sin A\sin B.

The sign pattern is worth noticing. For sine, the middle operation follows the sign between the angles: addition for \(A+B\), subtraction for \(A-B\). For cosine, the pattern is reversed: subtraction for \(A+B\), addition for \(A-B\).

A Reliable Calculation Method

First choose a decomposition that uses angles whose exact values are known. For example, \(75^\circ=45^\circ+30^\circ\), while \(15^\circ=45^\circ-30^\circ\). Then substitute the special-angle values into the correct formula. Do not decide whether the answer is positive or negative by looking only at the signs of the terms in the formula: check the quadrant of the complete angle.

Decompose the angle.
Write the target as a sum or difference of convenient special angles.

Select the formula.
Match the target function and the chosen sum or difference to the correct addition formula.

Substitute and simplify.
Insert exact special-angle values and combine the resulting fractions and radicals.

Check the result.
Confirm that its sign agrees with the target angle’s quadrant, and use a related identity or ratio when helpful.

Worked Examples

Worked Example: Find \(\sin 75^\circ\)

Write \(75^\circ=45^\circ+30^\circ\). Use the sine addition formula:

\begin{aligned} \sin 75^\circ &=\sin(45^\circ+30^\circ)\\ &=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ\\ &=\left(\frac{\sqrt2}{2}\right)\left(\frac{\sqrt3}{2}\right) +\left(\frac{\sqrt2}{2}\right)\left(\frac12\right)\\ &=\frac{\sqrt6}{4}+\frac{\sqrt2}{4}\\ &=\frac{\sqrt6+\sqrt2}{4}. \end{aligned}

This is an exact value. As a sign check, \(75^\circ\) lies in quadrant I, where sine is positive, and the result is positive. The calculation also provides a derivation: it follows by substituting the special-angle values into the sine addition formula and simplifying.

For an additional check, the same formula for cosine gives \(\cos75^\circ=(\sqrt6-\sqrt2)/4\). These two values satisfy the Pythagorean identity:

\left(\frac{\sqrt6+\sqrt2}{4}\right)^2 +\left(\frac{\sqrt6-\sqrt2}{4}\right)^2 = \frac{8+4\sqrt3}{16}+\frac{8-4\sqrt3}{16} =1.

Thus the paired sine and cosine values are consistent with \(\sin^2 75^\circ+\cos^2 75^\circ=1\).

Worked Example: Find \(\cos 15^\circ\)

Write \(15^\circ=45^\circ-30^\circ\). The cosine difference formula has a plus sign between its two products:

\begin{aligned} \cos15^\circ &=\cos(45^\circ-30^\circ)\\ &=\cos45^\circ\cos30^\circ+\sin45^\circ\sin30^\circ\\ &=\left(\frac{\sqrt2}{2}\right)\left(\frac{\sqrt3}{2}\right) +\left(\frac{\sqrt2}{2}\right)\left(\frac12\right)\\ &=\frac{\sqrt6}{4}+\frac{\sqrt2}{4}\\ &=\frac{\sqrt6+\sqrt2}{4}. \end{aligned}

The result is positive, as expected because \(15^\circ\) is in quadrant I. Notice that \(\cos15^\circ\) equals \(\sin75^\circ\); this is consistent with the complementary-angle identity because \(15^\circ+75^\circ=90^\circ\).

We can also check the value by squaring it:

\left(\frac{\sqrt6+\sqrt2}{4}\right)^2 =\frac{8+4\sqrt3}{16} =\frac{2+\sqrt3}{4}.

This square is positive and less than \(1\), as required for the square of a cosine value.

Worked Example: Find \(\cos 105^\circ\)

Use \(105^\circ=60^\circ+45^\circ\). The cosine addition formula is:

\begin{aligned} \cos105^\circ &=\cos(60^\circ+45^\circ)\\ &=\cos60^\circ\cos45^\circ-\sin60^\circ\sin45^\circ\\ &=\left(\frac12\right)\left(\frac{\sqrt2}{2}\right) -\left(\frac{\sqrt3}{2}\right)\left(\frac{\sqrt2}{2}\right)\\ &=\frac{\sqrt2}{4}-\frac{\sqrt6}{4}\\ &=\frac{\sqrt2-\sqrt6}{4}. \end{aligned}

Since \(\sqrt6>\sqrt2\), the numerator is negative. This agrees with the quadrant check: \(105^\circ\) lies in quadrant II, where cosine is negative. Reversing the subtraction in the final numerator would give a value with the wrong sign.

As a consistency check, the answer’s magnitude is less than \(1\). Indeed, \(\sqrt6-\sqrt2\) is positive and less than \(4\), so \(\left|\cos105^\circ\right|=(\sqrt6-\sqrt2)/4<1\), as cosine values must be.

Worked Example: Find \(\tan 15^\circ\)

Tangent is the quotient of sine and cosine where cosine is nonzero. We can obtain both values using \(15^\circ=45^\circ-30^\circ\):

\begin{aligned} \sin15^\circ &=\sin45^\circ\cos30^\circ-\cos45^\circ\sin30^\circ\\ &=\left(\frac{\sqrt2}{2}\right)\left(\frac{\sqrt3}{2}\right) -\left(\frac{\sqrt2}{2}\right)\left(\frac12\right)\\ &=\frac{\sqrt6-\sqrt2}{4}, \end{aligned}

\cos15^\circ=\frac{\sqrt6+\sqrt2}{4}.

Because \(\cos15^\circ\ne0\), divide the two exact values and simplify:

\begin{aligned} \tan15^\circ &=\frac{\sin15^\circ}{\cos15^\circ}\\ &=\frac{\sqrt6-\sqrt2}{\sqrt6+\sqrt2}\\ &=\frac{(\sqrt6-\sqrt2)^2}{(\sqrt6+\sqrt2)(\sqrt6-\sqrt2)}\\ &=\frac{8-4\sqrt3}{6-2}\\ &=\frac{8-4\sqrt3}{4}\\ &=2-\sqrt3. \end{aligned}

The rationalizing step multiplies numerator and denominator by \(\sqrt6-\sqrt2\), which is nonzero. The answer is positive, as tangent is positive in quadrant I. Substituting the values back into the quotient gives \((\sqrt6-\sqrt2)/(\sqrt6+\sqrt2)=2-\sqrt3\), confirming the simplification.

Why the Method Works

The angle addition formulas express a trigonometric value at a combined angle using products of values at its component angles. If the component angles are special angles, each factor is known exactly. The calculation therefore reduces to arithmetic with fractions and square roots; no decimal approximation is needed.

For example, the derivation of \(\sin75^\circ\) used only \(75^\circ=45^\circ+30^\circ\), the sine addition formula, and the exact values at \(45^\circ\) and \(30^\circ\). Each substitution preserves equality, and collecting the two terms over a common denominator yields \((\sqrt6+\sqrt2)/4\). This explains why the method gives an exact result rather than merely an estimate.

Signs and a Common Mistake

The quadrant of the target angle is a separate check from the formula’s algebra. In quadrant I, sine and cosine are positive; in quadrant II, sine is positive and cosine negative. For tangent, the sign is positive in quadrants I and III and negative in quadrants II and IV, wherever tangent is defined. Use the target angle’s location, not just a visual impression of one term in the formula.

A frequent mistake is to use the sum formula for cosine when the angle is a difference. For instance, an incorrect calculation of \(\cos(45^\circ-30^\circ)\) might be:

\cos(45^\circ-30^\circ) \ne \cos45^\circ\cos30^\circ-\sin45^\circ\sin30^\circ =\frac{\sqrt6-\sqrt2}{4}.

That subtraction belongs to the cosine sum formula, not the difference formula. The correct calculation is:

\begin{aligned} \cos(45^\circ-30^\circ) &=\cos45^\circ\cos30^\circ+\sin45^\circ\sin30^\circ\\ &=\frac{\sqrt6+\sqrt2}{4}. \end{aligned}

The incorrect result is actually \(\sin15^\circ\), not \(\cos15^\circ\). Both values happen to be positive, so checking only the quadrant would not catch this error. Matching the formula to the operation between the angles is essential.

Choosing a Decomposition

A target angle may have more than one useful representation. Prefer one that uses special angles and keeps the arithmetic simple. For example, \(75^\circ\) can be written as \(45^\circ+30^\circ\), while \(15^\circ\) can be written as \(45^\circ-30^\circ\). After choosing, keep that decomposition visible so the signs in the formula remain clear.

When exact values are requested, leave the answer in exact form. Do not replace \(\sqrt2\) or \(\sqrt3\) with rounded decimals. If the result involves a quotient of radicals, simplify it exactly when possible, as in the tangent example. Finally, check that the result’s sign and size make sense for the function and angle.

Takeaway: Decompose the angle, use the matching addition formula, substitute exact special-angle values, simplify carefully, and check the final sign using the target angle’s quadrant.

Check Your Understanding

For each question, show the formula used and keep your final answer exact.

Write \(105^\circ\) as a sum of special angles and use the sine addition formula to find \(\sin105^\circ\).
Use a difference of special angles to find \(\sin15^\circ\).
Find \(\cos75^\circ\), and explain why its sign is consistent with the quadrant of \(75^\circ\).
Find \(\tan75^\circ\) by dividing your exact values for \(\sin75^\circ\) and \(\cos75^\circ\).
State the correct sign between the two products in the formula for \(\cos(A-B)\), and identify the common error it prevents.

Exact Values from Angle Addition Formulas

What You'll Learn

From Special Angles to New Exact Values

Special-Angle Values and Addition Formulas

A Reliable Calculation Method