Use the given information to find ( a ) sin(s+t),(b) tan(s+t), and (c) the quadrant of s+t . sin

Use the given information to find ( a ) sin(s+t),(b)...

Key Concepts

Sine Addition Formula

This formula expresses the sine of the sum of two angles in terms of the sines and cosines of the individual angles: sin(s+t) = sin s · cos t + cos s · sin t. It is essential in solving problems where the sine of a summed angle is required, especially when the individual trigonometric values are known.

Pythagorean Identity

The Pythagorean Identity relates the sine and cosine of an angle by the equation sin^2? + cos^2? = 1. This identity is used to determine the cosine (or sine) of an angle when the sine (or cosine) is known. The sign of the resulting cosine depends on the quadrant in which the angle lies.

Tangent Addition Formula

The tangent addition formula allows you to calculate the tangent of a sum of two angles using the individual tangent values: tan(s+t) = (tan s + tan t) / (1 - tan s · tan t). This formula is particularly useful when the individual tangents are easier to compute or are given, and the sum's tangent is required.

Trigonometric Ratios in Quadrants

Understanding the signs of trigonometric ratios in different quadrants is crucial. For instance, knowing that sine and cosine have specific positive or negative signs depending on the quadrant helps determine the correct value when applying identities like the Pythagorean Identity or when discerning the quadrant of an angle sum.

Quadrant Determination for Angle Sums

Determining the quadrant in which an angle (or sum of angles) lies is important for assigning the proper sign to trigonometric values. It involves analyzing the sum of the individual angles and using the signs of sine and cosine to establish the overall angle's quadrant.

Transcript

00:01 In order to find the sign of s plus t, knowing the sign of s and the sign of t, we first need to find the cosine of s and the cosine of t.

00:12 Since we know that sine or cosine squared of theta is equal to 1 minus sine squared of theta, we can find cosine squared of theta by doing 1 minus sine squared of theta.

00:31 Our sign squared.

00:34 So we're going to use s first.

00:38 So to find the cosine squared of s, we can do 1 minus the sign of s squared, which is 4 over 9.

00:49 So that tells us the cosine squared of s is 5 over 9, which tells us that cosine of s is the square root of 5 over 3.

01:08 We can use the same logic to the cosine of t.

01:13 So the cosine squared of t is equal to 1 minus the sine squared of t which is 1 minus negative 1 3rd squared and 1 3rd squared would be 1 over 9 so cosine squared of t is equal to 8 9th so that tells us that the cosine of t is equal to the square root of 8, which is 2 root 2 over 3.

02:05 Now when working with square roots, we can have the positive and the negative version.

02:11 So we need to check our quadrants.

02:14 Since s is in quadrant 2, and that is where sign is positive but cosine is negative, that tells us that the cosine of, s negative square root of 5 over 3 and the t is in quadrant 4 which is where cosine is positive so our cosine of t will be positive 2 or 2 over 3 right so now we can use those values to help us to find our sign of s plus t so the sign of s plus t is equal to the sign of s plus t is equal to the sign of s times the cosine of t, plus the cosine of s times the sign of t.

03:09 So we can replace the values we have to the right.

03:13 The sign of s is two thirds, and the cosine of t was two root two over three, plus the cosine of s, which is negative root five over three, times the sign of t, which is negative root five negative one -third.

03:38 So this gives us 4 root 2 over 3 plus, because a negative and a negative makes it positive.

03:51 Sorry, not over 3, over 9.

03:55 4 root 2 over 9 plus root 5 over 9...

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