Finding sin t cos t from sin t + cos t

We are given the equation: $$\sin t+\cos t=\frac{4}{5}$$ Our goal is to find the value of the product sin t cos t.

Using Trigonometric Identities to Solve

To find sin t cos t from the sum sin t + cos t, we can use the algebraic identity for squaring a binomial and a fundamental trigonometric identity.

Consider squaring the given equation:

$${(\sin t+\cos t)}^2={\left(\frac{4}{5}\right)}^2$$

Expand the left side using the formula ${(a+b)}^2=a^2+2ab+b^2$:

$${\sin }^{2}t+2\sin t\cos t+{\cos }^{2}t=\frac{16}{25}$$

Rearrange the terms on the left side:

$$({\sin }^{2}t+{\cos }^{2}t)+2\sin t\cos t=\frac{16}{25}$$

We know the fundamental trigonometric identity: $${\sin }^{2}t+{\cos }^{2}t=1$$

Substitute this identity into the equation:

$$1+2\sin t\cos t=\frac{16}{25}$$

Now, we need to isolate the term $2\sin t\cos t$. Subtract 1 from both sides of the equation:

$$2\sin t\cos t=\frac{16}{25}-1$$

To subtract 1, express 1 as a fraction with denominator 25: $1=\frac{25}{25}$.

$$2\sin t\cos t=\frac{16}{25}-\frac{25}{25}$$

Perform the subtraction:

$$2\sin t\cos t=\frac{16-25}{25}$$

$$2\sin t\cos t=\frac{-9}{25}$$

Finally, divide both sides by 2 to find sin t cos t:

$$\sin t\cos t=\frac{\left(\frac{-9}{25}\right)}{2}$$

Dividing by 2 is the same as multiplying by $\frac{1}{2}$:

$$\sin t\cos t=\frac{-9}{25}\times \frac{1}{2}$$

$$\sin t\cos t=\frac{-9}{50}$$

Conclusion

Given $\sin t+\cos t=\frac{4}{5}$, we found that $\sin t\cos t=\frac{-9}{50}$.

Revision Table: Key Trigonometric Concepts

Additional Information: Solving Trigonometric Equations

Problems involving sums or differences of sin t and cos t can often be solved by squaring the expression. This introduces the term $2\sin t\cos t$, which is related to $\sin \left(2t\right)$ by the double angle identity .

The fact that sin t + cos t is positive (4/5) does not restrict sin t cos t to be positive. As seen in this problem, the product sin t cos t can be negative. This typically happens when t is in a quadrant where sin t and cos t have opposite signs (Quadrant II or IV).

For example, if t is in Quadrant II, sin t > 0 and cos t < 0, so sin t cos t < 0. If t is in Quadrant IV, sin t < 0 and cos t > 0, so sin t cos t < 0.