Calculate the value of the following, without using a calculator, leaving answers, where necessary, in simplified surd form:
(a). sin 120° + tan 300°
(b). tan 315° × cos330° / sin(-240°) × sin 570°
I need your response
You must have a list of simple trig values for the 30-60-90 and the 45-45-90 right-angled triangles
Make yourself a sketch and keep it handy, the corresponding sides would be
1-√3-2 and 1-1-√2
eg. sin30° = opposite/hypotenuse = 1/2
This in conjunction with the CAST rule will allow you to find trig values of any angle which is a combination of 180°, 360° and the above angles
e.g. sin120° = sin60° , (120 is in quadrant II and in II the sine is positive, and 120 is 60° from 180 <----- the x-axis)
and sin 60 = √3/2, so sin120° = √3/2
tan 300° = -√3/1 = -√3, because 300° is in IV and tangent is negative, and 300° is 60° from the x-axis
so sin 120° + tan 300° = √3/2 + (-√3) = -√3/2
for the 2nd part, notice 315 = 360-45, 330 = 360-30
-240° = coterminal with +120° = 180-60
570 = 360° + 210°
and 210 = 180 + 30
let me know what you get for part 2
To calculate the value of the given expressions without using a calculator, we can use the unit circle and trigonometric identities. Let's break down each expression step by step:
(a) sin 120° + tan 300°:
1. Start with sin 120°. On the unit circle, 120° corresponds to the point (-1/2, √3/2). Therefore, sin 120° = √3/2.
2. Next, let's calculate tan 300°. Using the identity tan θ = sin θ / cos θ, we can rewrite tan 300° as sin 300° / cos 300°.
3. On the unit circle, 300° corresponds to the point (-√3/2, -1/2). Therefore, sin 300° = -√3/2 and cos 300° = -1/2.
4. Plugging these values into tan 300°, we get (-√3/2) / (-1/2) = √3.
Now, we can add sin 120° and tan 300°:
√3/2 + √3
Since the denominators are the same, we can combine the terms:
(√3 + 2√3) / 2
= 3√3 / 2
Therefore, the value of sin 120° + tan 300° is 3√3 / 2.
(b) tan 315° × cos 330° / sin(-240°) × sin 570°:
Let's break down each part of the expression:
1. tan 315°:
On the unit circle, 315° corresponds to the point (-√2/2, -√2/2). Therefore, tan 315° = -√2.
2. cos 330°:
On the unit circle, 330° corresponds to the point (√3/2, -1/2). Therefore, cos 330° = √3/2.
3. sin(-240°):
The sine function is an odd function, which means sin(-θ) = -sin(θ). Therefore, sin(-240°) = -sin(240°).
On the unit circle, 240° corresponds to the point (-√3/2, 1/2). Therefore, sin 240° = 1/2. Therefore, sin(-240°) = -1/2.
4. sin 570°:
Since we know sin 570°, we don't need to calculate it again. It is the same as sin 210°, which is -1/2 based on the unit circle.
Now, let's substitute the values into the expression:
(-√2) × (√3/2) / (-1/2) × (-1/2)
Simplifying each part:
= -√6 / 2 / 1/4
= -√6 / 2 * 4/1
= -2√6
Therefore, the value of tan 315° × cos 330° / sin(-240°) × sin 570° is -2√6.
In both cases, we obtained the values without using a calculator by using the unit circle and trigonometric identities.